Given the graph of in the figure, determine all x-values from F, G, H, or I for which the function is differentiable.
A)F G H I
B)F H
C)G H I
D)F G
E)F H I

Question

Given the graph of  in the figure, determine all x-values from F, G, H, or I for which the function is differentiable. ​  ​
A)F G H I 
B)F H 
C)G H I 
D)F G 
E)F H I

Quizplus · Accepted Answer

Differentiability requires that the graph of the function has a well-defined tangent line at that point. We observe that at points F and I, the graph has a sharp corner or cusp, which means that the tangent line is not well-defined at these points, and hence the function is not differentiable there. At point G, the graph has a vertical tangent, which also means that the tangent line is not well-defined, and hence the function is not differentiable at that point. On the other hand, at point H, the graph has a well-defined tangent line, and hence the function is differentiable at that point. Therefore, the x-values for which the function is differentiable are G, H, and I. Hence, the answer is C.

Given the Graph of in the Figure, Determine All