How do you figure out critical points and anti-derivatives?

Question

Consider the function f(x) = ex – kx, where k is a positive constant and x ≥ 0.

a. how do u Find all the critical points of f in terms of k.
b. how do u Find the anti-derivative F(x) satisfying F(0) = 3.

Answer 1

Did you mean e^x?

If so, then f' = 0 means e^x - k = 0. I.e., x = ln k is the only critical point.

The anti-derivatives are e^x - kx^2/2 + constant. At 0 they just equal 1 + that constant. So you're looking for the one with constant = 2. :)