Question

sketch the graph of f and determine where f is nondifferentiable.
f(x)=\begin{cases}-2x & \text{if }x < 1\\-2 & \text{if }xgeq1end{cases}
a.
b.
c.
d.
where is the function f(x) nondifferentiable? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function f(x) is nondifferentiable at x =

b. the function f(x) is differentiable for all real numbers.

Explanation:

Step1: Analyze left - hand limit of derivative

For \(x < 1\), \(f(x)=-2x\). The derivative \(f^\prime(x)=-2\).

Step2: Analyze right - hand limit of derivative

For \(x\geq1\), \(f(x)= - 2\). The derivative \(f^\prime(x)=0\).

Step3: Check differentiability at \(x = 1\)

The left - hand derivative \(f^\prime_{-}(1)=-2\) and the right - hand derivative \(f^\prime_{+}(1)=0\). Since \(f^\prime_{-}(1)
eq f^\prime_{+}(1)\), the function is non - differentiable at \(x = 1\).

Answer:

A. The function \(f(x)\) is nondifferentiable at \(x = 1\)