For the piecewise function: f(x) = ax^2 where d > x; f(x) = c * log(x) - 2 where d < x. What values do a and c have to take on such that f is differentiable at x = d > 0?

Question

1. a, c, and d all belong to real numbers
2. the answer should be in terms of d

I have gotten to an answer but am completely unsure if I have done it correctly. I would like some insight as to how to solve this question

Answer 1

In order for the piecewise function to be differentiable at d:

1) It must be continuous at d meaning f(d) for the left function must = f(d) for the right function.

2) The derivative at f(d) for the left function must = the derivative at f(d) for the right function.

For 1) a(d²) = c*log(d) - 2 (Note, it is unclear to me if log(d) represents log base 10 of d or ln(d) so I won't proceed with the details beyond this.

For 2) f'(x) for the left function = 2ax so f'(d) = 2ad and for the right function f'(x) = c/(x) or c/(xln10) depending on what the base of the logarithm is so f'(d) = c/d or c/(dln10) depending on the base of the log. Set those equal to get the second equation and solve for a and c in terms of d.