Question

the graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points (1,-1) and (3,1). for what values of x, -2<x<4, is f not differentiable?

Explanation:

Step1: Recall the condition for non - differentiability

A function \(y = f(x)\) is not differentiable at a point \(x=a\) if there is a discontinuity, a corner (a sharp turn), or a vertical tangent at \(x = a\).

Step2: Analyze the given graph

• A vertical tangent occurs when the slope of the tangent line approaches \(\pm\infty\).
• A function is differentiable if the left - hand derivative and the right - hand derivative are equal.
• For the given function \(y = f(x)\) with \(-2\lt x\lt4\):
• At \(x=-1\), the function has a corner (a sharp turn). The left - hand derivative and the right - hand derivative are not equal.
• At \(x = 0\), the function has a discontinuity (a break in the graph).
• At \(x=2\), the function has a vertical tangent. The slope of the tangent line is undefined (approaches \(\pm\infty\)).

Answer:

\(x=-1,x = 0,x=2\)