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AP Calculus BC : Second Derivatives

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Example Questions

Example Question #21 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=x^{2} - 6x + 2x^{3} - 2x^{4} + 6\end{align*}$

Possible Answers:

-0.264,2.528

-0.132,0.632

-0.923,3.159

$\text{There are no points of inflection.}$

Correct answer:

-0.132,0.632

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&2x + 6x^{2} - 8x^{3} - 6\\&\text{And our second is:}\\&12x - 24x^{2} + 2\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.132,0.632\\&\text{We find real unique roots at the points }x=-0.132,0.632\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #22 : Second Derivatives

$\begin{align*}&\text{Determine what, if any exist, the points of inflection are for the function}\\&y=5x^{3} - 10x^{2} - 10x - 5\end{align*}$

Possible Answers:

0.667

$\text{There are no points of inflection.}$

4.667

2.000

Correct answer:

0.667

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&15x^{2} - 20x - 10\\&\text{And our second is:}\\&30x - 20\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.667\\&\text{We find one real unique root at the point }x=0.667\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #23 : Second Derivatives

$\begin{align*}&\text{Find any points of inflection for the function:}\\&7x - 2x^{2} - 2x^{3} + 6\end{align*}$

Possible Answers:

-1.333

$\text{There are no points of inflection.}$

-0.333

-2.000

Correct answer:

-0.333

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&7 - 6x^{2} - 4x\\&\text{And our second is:}\\&- 12x - 4\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.333\\&\text{We find one real unique root at the point }x=-0.333\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #24 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=8x - 9x^{2} + 3x^{3} + 8x^{4} + 8x^{5} + 8\end{align*}$

Possible Answers:

0.297

$\text{There are no points of inflection.}$

2.079

1.188

Correct answer:

0.297

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&9x^{2} - 18x + 32x^{3} + 40x^{4} + 8\\&\text{And our second is:}\\&18x + 96x^{2} + 160x^{3} - 18\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.297,-0.448+0.422i,-0.448-0.422i\\&\text{We find one real unique root at the point }x=0.297\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #25 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=2x^{3} - 3x^{2} - 2x^{4} - 8x^{5} - 1\end{align*}$

Possible Answers:

-1.424

-0.475

-3.322

$\text{There are no points of inflection.}$

Correct answer:

-0.475

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&6x^{2} - 6x - 8x^{3} - 40x^{4}\\&\text{And our second is:}\\&12x - 24x^{2} - 160x^{3} - 6\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.475,0.162-0.230i,0.162+0.230i\\&\text{We find one real unique root at the point }x=-0.475\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #26 : Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=3x + 8x^{2} - 3x^{3} - 10x^{4} - 5\end{align*}$

Possible Answers:

-0.448,0.298

$\text{There are no points of inflection.}$

-0.896,1.191

-3.134,2.084

Correct answer:

-0.448,0.298

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&16x - 9x^{2} - 40x^{3} + 3\\&\text{And our second is:}\\&16 - 120x^{2} - 18x\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.448,0.298\\&\text{We find real unique roots at the points }x=-0.448,0.298\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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Example Question #26 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=\frac{1}{2^{(3x)}}- x - 10x^{2} + x^{3} - 10x^{4} + 6x^{5} + 3\end{align*}$

Possible Answers:

2.182

7.637

$\text{There are no points of inflection.}$

1.091

Correct answer:

1.091

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&3x^{2} -\frac{ (3ln(2))}{2^{(3x)}}- 20x - 40x^{3} + 30x^{4} - 1\\&\text{And our second is:}\\&6x +\frac{ (9ln(2)^{2})}{2^{(3x)}}- 120x^{2} + 120x^{3} - 20\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=1.091\\&\text{We find one real unique root at the point }x=1.091\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #28 : Derivatives

$\begin{align*}&\text{Find any points of inflection for the function:}\\&3^{(3x)} - 4x + 10x^{3} + 10x^{4} + 2x^{5}\end{align*}$

Possible Answers:

-0.026

-0.154

-0.077

$\text{There are no points of inflection.}$

Correct answer:

-0.154

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&3\cdot 3^{(3x)}ln(3) + 30x^{2} + 40x^{3} + 10x^{4} - 4\\&\text{And our second is:}\\&60x + 9\cdot 3^{(3x)}ln(3)^{2} + 120x^{2} + 40x^{3}\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.154\\&\text{We find one real unique root at the point }x=-0.154\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #29 : Derivatives

$\begin{align*}&\text{Find any points of inflection for the function:}\\&e^{(2x)} - 8x + 8x^{2} + 5x^{3} - 2x^{4} - 7x^{5} - 2\end{align*}$

Possible Answers:

0.132

$\text{There are no points of inflection.}$

0.660

0.330

Correct answer:

0.660

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&16x + 2e^{(2x)} + 15x^{2} - 8x^{3} - 35x^{4} - 8\\&\text{And our second is:}\\&30x + 4e^{(2x)} - 24x^{2} - 140x^{3} + 16\\&\text{For a reminder of derivative rules:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=0.660\\&\text{We find one real unique root at the point }x=0.660\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

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Example Question #26 : Inflection Points

$\begin{align*}&\text{Find any points of inflection for the function:}\\&4x + ln(5x) - 6x^{2} - 10x^{3} + 6x^{4} - 5\end{align*}$

Possible Answers:

-0.292,1.012

$\text{There are no points of inflection.}$

-1.462,5.058

-0.877,0.506

Correct answer:

-0.292,1.012

Explanation:

$\begin{align*}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.292,1.012,0.057-0.209i,0.057+0.209i\\&\text{We find real unique roots at the points }x=-0.292,1.012\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

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AP Calculus BC : Second Derivatives

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #21 : Second Derivatives

Example Question #22 : Second Derivatives

Example Question #23 : Second Derivatives

Example Question #24 : Second Derivatives

Example Question #25 : Second Derivatives

Example Question #26 : Derivatives

Example Question #26 : Second Derivatives

Example Question #28 : Derivatives

Example Question #29 : Derivatives

Example Question #26 : Inflection Points

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