Q. 3 - Calculus

Question

If f has a local maximum at x = 1, then what can you say about f '(1)?  What if you also know that f is differentiable at x = 1 .

Step-by-Step Solution

Answer

If the function f is differentiable at x = 1 then f'(1) must be equal to zero .

Step 1. Given information .

Consider the function If f has a local maximum at x = 1 .

Step 2. Find the function.

The function $f\left(x\right)=x-1$has root $x=1$at this point the function f has local maximum . Differentiate the given function .

$$\begin{gathered} f\left(x\right)=x-1 \\ f\text{'}\left(x\right)=1 \\ f\text{'}\left(1\right)=1 \end{gathered}$$

The derivative of the given function is not equal to zero so the function is differentiable at $x=1$ but not continuous and not satisfied the condition $f\text{'}\left(1\right)=0$ .