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

Question

f(x) is decreasing at x= -6

f(x) has a local minimum at x= -3

f(x) has a local maximum at x= 3

Answer 1

The simple solution is -(x+3)^2 x (x-3)^2

Note that this function will be 0 at -3, 0, and 3, positive on (-∞, -3) U (-3, 0), and negative on (0, 3) U (3, ∞), going to ∞ as x goes to -∞ and going to -∞ as x goes to ∞

Also, see that decreasing at x = -6 means a negative slope. When we see a graph, we will think of it as increasing to positive infinity as x goes to - infinity, but that means decreasing.