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Precalculus : Determine Points of Inflection

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Example Question #132 : Derivatives

Find the points of inflection for the following function:

$\displaystyle f(x)=x^2-16x+100$

Possible Answers:

$\displaystyle 4$

$\displaystyle 16$

100 $\displaystyle 100$

none $\displaystyle none$

Correct answer:

none $\displaystyle none$

Explanation:

To solve, find when the 2nd derivative is equal to 0 and plug it into the first equation. We must use the power rule, as outlined below, to find each derivative.

Power rule: $(x^n)'=nx^{n-1}$ $\displaystyle (x^n)'=nx^{n-1}$

Thus,

$\displaystyle f(x)=x^2-16x+100$

$f'(x)=2x^{2-1}-16x^{1-1}=2x-16x^0=2x-16$ $\displaystyle f'(x)=2x^{2-1}-16x^{1-1}=2x-16x^0=2x-16$

$f''(x)=1*2x^{1-1}=2x^0=2$ $\displaystyle f''(x)=1*2x^{1-1}=2x^0=2$

Since 2 is never equal to 0, there are no points of inflection.

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Example Question #21 : Determine Points Of Inflection

Determine the $\displaystyle x$ values for the points of inflection of the following function:

$\displaystyle f(x)=x^3-x^2+x-1$

Possible Answers:

$\displaystyle -2$

$\displaystyle 3$

$\frac{-1}{3}$ $\displaystyle \frac{-1}{3}$

$\frac{1}{3}$ $\displaystyle \frac{1}{3}$

Correct answer:

$\frac{1}{3}$ $\displaystyle \frac{1}{3}$

Explanation:

To solve, you must set the second derivative equal to 0 and solve for x. To differentiate twice, use the power rule as outlined below:

Power Rule: $(x^n)'=nx^{n-1}$ $\displaystyle (x^n)'=nx^{n-1}$

Therefore:

$f'(x)=3x^{3-1}-2x^{2-1}+x^{1-1}=3x^2-2x$ $\displaystyle f'(x)=3x^{3-1}-2x^{2-1}+x^{1-1}=3x^2-2x$

Remember, the derivative a constant is 0.

$f''(x)=3*2x^{2-1}-2x^{1-1}=6x-2$ $\displaystyle f''(x)=3*2x^{2-1}-2x^{1-1}=6x-2$

Now, set it equal to 0. Thus,

6x-2=0 $\displaystyle 6x-2=0$

6x=2 $\displaystyle 6x=2$

$x=\frac{2}{6}$ $\displaystyle x=\frac{2}{6}$

$x=\frac{1}{3}$ $\displaystyle x=\frac{1}{3}$

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Precalculus : Determine Points of Inflection

Study concepts, example questions & explanations for Precalculus

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Example Question #132 : Derivatives

Example Question #21 : Determine Points Of Inflection

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