how would we start this

Question

generate a continuous and differentiable function f(x)fx with the following properties:

1. f(x)fx is decreasing at x=−5x=-5
2. f(x)fx has a local minimum at x=−2x=-2
3. f(x)fx has a local maximum at x=2x=2

 

Answer 1

Let investigate the first derivative f '(x).

f(x) is decreasing at x = -5, so (a) f '(-5) < 0

f(x) has a local minimum at x = -2, so (b) f '(-2) = 0 and f '(x) changes from negative to positive at x = -2.

f(x) has a local maximum at x = 2, so (c) f '(2) = 0 and f '(x) changes from positive to negative at x = 2.

Generate a continuous function f '(x) which has the properties (a), (b), and (c).

The easiest such function is the parabola 4 - x²

So, we can take f '(x) = 4 - x²

Take the antiderivative: f(x) = 4x - x³/3