Question

determine if the following piecewise defined function is differentiable at x = 0.
f(x)=\begin{cases}3x - 5, & xgeq0\\x^{2}+4x - 5, & x < 0end{cases}
what is the right - hand derivative of the given function?
lim_{h
ightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=3 \text{ (type an integer or a simplified fraction.)}
what is the left - hand derivative of the given function?
lim_{h
ightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=4 \text{ (type an integer or a simplified fraction.)}
is the given function differentiable at x = 0?
no
yes

Explanation:

Step1: Recall differentiability condition

A function is differentiable at a point if the left - hand derivative and the right - hand derivative at that point are equal.

Step2: Identify left - hand and right - hand derivatives

The right - hand derivative $\lim_{h
ightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=3$ and the left - hand derivative $\lim_{h
ightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=4$.

Step3: Compare the derivatives

Since $3
eq4$, the left - hand derivative and the right - hand derivative are not equal at $x = 0$.

Answer:

No