Question

A function $f$ has a derivative for each $x$ such that $|x|<2$ and has a local minimum at (2,-5) . Which statement below must be true? (A) $f^{\prime}(2)=0$ (B) $f^{\prime}$ exists at $x=2$ (C) $f^{\prime}(x)<0$ if $x<2, f^{\prime}(x)>0$ if $x>2$ (D) none of the preceding is necessarily true

Name: Problem 41, Chapter :
Uploaded: 2021-11-24T15:12:44-08:00
Duration: 58 s
Channel: Hast Aggarwal
Description: A function f has a derivative for each x such that |x| 0 if x>2 (D) none of the preceding is necessarily true

   A function $f$ has a derivative for each $x$ such that $|x|<2$ and has a local minimum at (2,-5) . Which statement below must be true?
(A) $f^{\prime}(2)=0$
(B) $f^{\prime}$ exists at $x=2$
(C) $f^{\prime}(x)<0$ if $x<2, f^{\prime}(x)>0$ if $x>2$
(D) none of the preceding is necessarily true

Instant Answer

Solved by Expert Hast Aggarwal

11/24/2021

Step 1

In other words, the derivative of the function, $f'(x)$, should change from negative to positive as we move from left to right across $x=2$. (A) $f^{\prime}(2)=0$: This statement is true. At a local minimum, the derivative of the function is 0, since the Show more…

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A function $f$ has a derivative for each $x$ such that $|x|<2$ and has a local minimum at (2,-5) . Which statement below must be true? (A) $f^{\prime}(2)=0$ (B) $f^{\prime}$ exists at $x=2$ (C) $f^{\prime}(x)<0$ if $x<2, f^{\prime}(x)>0$ if $x>2$ (D) none of the preceding is necessarily true

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Key Concepts

Local Extremum and Fermat’s Theorem

Fermat’s Theorem states that if a function has a local extremum at an interior point and is differentiable there, then its derivative must be zero. However, this theorem does not apply to endpoints of the domain, where the standard notion of the derivative as a two-sided limit may not exist.

Differentiability at Endpoints

At an endpoint of a function's domain, the derivative may only be defined from one side (a one-sided derivative) rather than both. This means that the usual conditions that guarantee a derivative of zero at local extrema do not necessarily hold at endpoints, making the sign behavior of the derivative or its existence at those points non-assumable.

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