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

Question

f(x) is decreasing at x=-5

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

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

Answer 1

The obvious choice would be an odd function which is periodic, which makes sin the obvious choice. Recall that sin has a max at π/2 and min at -π/2. We change this to 3 and -3 by changing the period appropriately.

sin((π/2)/3 x) = sin(π/6 x)

Note that, when x = 3, sin(π/6 x) = sin(π/6 * 3) = sin(π/2) = 1, the max of the sin function, and

when x = -3, sin(π/6 (-3)) = sin(-π/2) = -1, the minimum of the sin function.

If this were a trig class, I would simply note that, as this is the sin cycle, it is decreasing from -9 to -3, and -5 is in this interval. In Calculus, we might prefer to take the derivative, and sin(π/6 x)' = π/6 cos(π/6 x), so the derivative at -5 will be π/6 cos(π/6 (-5)) = π/6 cos(-5π/6) = π/6 cos(-5π/6) (cos is even) = π/6 cos(150°) = -√3π/12 is negative, or decreasing, as the question requires.

Now, if you want to have a polynomial instead, the obvious choice is -(x-3)²(x)(x+3)²

This function has zeroes at -3, 0, and 3, and is positive on (-∞, -3) and (-3, 0) and negative on (0, 3) and (3, ∞) ; the x term was introduced to switch from maxima to minima, and the - sign was introduced to give a maximum at 3. You can calculate a derivative if you must to see that the function is decreasing at -5, but the function is clearly monotone on (-∞, -3) ; in any polynomial with all real zeroes, the function is monotone on the interval < the smallest zero and the interval > the largest zero; in this case, the monotonicity guarantees that it is decreasing on (-∞, -3), as the function is 0 at -3 and positive on the interval, going to ∞ as x goes to -∞.

Answer 2

Since f is continuous and differentiable, the three properties give us three more properties

1. Since f'(5)<0 then f'(x)<0 for x<-3 and x>3
2. f'(3)=f'(-3)=0
3. f'(x)>0 for -3<x<3

So the equation f'(x)=-(x+3)(x-3) has all three properties. If we multiply out, f'(x) = -x²+9. So if we want to find f, we need to think of a function whose derivative is -x²+9. By working backwards through the derivative process, you can see that one such function is f(x)=-x³/3+9x