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Example Question #103 : Derivative At A Point
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Example Question #871 : Ap Calculus Ab
Which of the following functions has exactly one point of inflection?
For any function, a point of inflection occurs when . For a function to have exactly one point of inflection, it has to be of third order (i.e. have an
term as its highest order term). The function
only has one point of inflection when
.
A linear equation does not have any points of inflection, nor does a quadratic equation or exponential function.
A function with an order higher than three has multiple points of inflection.
Example Question #872 : Ap Calculus Ab
Over what interval is the graph of concave upward?
The concavity of a function is determined by its second derivative. Whenever the second derivative of a function is positive, the function is said to be concave upward.
We will find the first derivative of using the Chain Rule.
Next, we will differentiate the first derivative of by employing the Product Rule.
We must now determine where .
We can divide both sides of the equation by . Since
is positive no matter what power it is raised to, we are left with the much simpler inequality:
This is only true when is less than or greater than
. Thus, the interval over which
is concave upward is
.
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