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Chapter 4: Problem 8

Give a function that does not have an inflection point at a point where \(f^{\prime \prime}(x)=0\)

Short Answer

Expert verified
Question: Provide an example of a function that does not have an inflection point at a point where its second derivative is zero. Answer: An example of such a function is \(f(x) = x^4\). The function has its second derivative zero at \(x = 0\), but the curvature does not change at this point, and hence there is no inflection point.

Step by step solution

01

Find the first derivative of the function

Find the first derivative of the function: $$ f^{\prime}(x) = \frac{d(x^4)}{dx} = 4x^3 $$
02

Find the second derivative of the function

Find the second derivative of the function: $$ f^{\prime \prime}(x) = \frac{d(4x^3)}{dx} = 12x^2 $$
03

Determine the points where the second derivative is zero

Set the second derivative equal to zero and find the points: $$ 12x^2 = 0 \Rightarrow x = 0 $$
04

Determine if there is an inflection point at \(x=0\)

We must analyze the sign of the second derivative around this point: For \(x < 0\), we have \(12x^2 > 0\). For \(x > 0\), we have \(12x^2 > 0\). Since the second derivative is positive on both sides of the point \(x = 0\), the function remains concave up and does not change its curvature. Therefore, \(f(x) = x^4\) does not have an inflection point at a point where its second derivative is zero.

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