Question

A continuous function f is defined on the closed interval −4 ≤ x ≤ 6. The graph of f consists of a line segment and a curve that is tangent to the x-axis at x = 3, as shown in the figure below. On the interval 0 < x < 6, the function f is twice differentiable, with f ′′( x) > 0.

Is f differentiable at x = 0 ? Use the definition of the derivative with one-sided limits to justify your answer.