the graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and…

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?

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?

Answer

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)