Final answer:
To make a piecewise function differentiable, the constants in the function must be chosen so that the function is continuous and has matching slopes at the points of intersection.
Step-by-step explanation:
To make a piecewise function differentiable, the constants in the function must be chosen in such a way that the function is continuous and has matching slopes at the points where the pieces are joined. This means that the value of the function and its derivative must match at the points of intersection between the pieces. Let's take an example:
Consider the piecewise function:
f(x) = { x², if x ≤ a;
2x - c, if x > a }
To make this function differentiable, we need to choose constants 'a' and 'c' such that the function is continuous and has matching slopes at the point x = a. This means that f(a) = a² and f'(a) = 2.