Question

consider the following function. (if an answer does not exist, enter dne.)$f(x) = |x - 7|$find the derivative from the left at $x = 7$.find the derivative from the right at $x = 7$.is the function differentiable at $x = 7$?○ yes○ no

Explanation:

Step1: Rewrite f(x) piecewise

For $x \geq 7$, $f(x) = x - 7$; for $x < 7$, $f(x) = 7 - x$

Step2: Left derivative at x=7

Use left limit definition:
$\lim_{h \to 0^-} \frac{f(7+h)-f(7)}{h} = \lim_{h \to 0^-} \frac{(7-(7+h)) - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$

Step3: Right derivative at x=7

Use right limit definition:
$\lim_{h \to 0^+} \frac{f(7+h)-f(7)}{h} = \lim_{h \to 0^+} \frac{(7+h-7) - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$

Step4: Check differentiability at x=7

A function is differentiable at a point if left and right derivatives are equal. Here, $-1
eq 1$.

Answer:

Derivative from the left at $x=7$: $-1$
Derivative from the right at $x=7$: $1$
Is the function differentiable at $x=7$? $\text{No}$