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

A function is differentiable at a point if and only if the tangent line at that point exists and is unique (i.e. the derivative exists). This means that the function must be continuous at that point as well. 
From the graph, we can see that the function is continuous and has a well-defined tangent line at all points on the interval FG, GH, and HI. Therefore, the function is differentiable at all points within these intervals. F, however, is a corner point where the tangent line cannot exist and thus the function at the point F is not differentiable. 
Therefore, the answer is C, since the function is differentiable only at points G, H, and I.

Given the Graph of in the Figure, Determine All