Question

generate a continuous and differentiable function f(x) with the following properties: f(x) is decreasing at x=−5 f(x) has a local minimum at x=−3 f(x) has a local maximum at x=3

Answer 1

Answer:

see details in graph and below

Step-by-step explanation:

There are many ways to generate the function.

We'll generate a function whose first derivative f'(x) satisfies the required conditions, say, a quadratic.

1. f(x) has a local minimum at x = -3, and

2. a local maximum at x = 3

Therefore f'(x) has to cross the x-axis at x = -3 and x=+3.

Furthermore, f'(x) must be increasing at x=-3 and decreasing at x=+3.

f'(x) = -x^2+9

will satisfy the above conditions.

Finally f(x) must be decreasing at x= -5, which implies that f'(-5) must be negative.

Check: f'(-5) = -(-5)^2+9 = -25+9 = -16 < 0  so ok.

f(x) can then be obtained by integrating f'(x) :

f(x) = integral of -x^2+9 = -x^3/3 + 9x = 9x - x^3/3

A graph of f(x) is attached, and is found to satisfy all three conditions.