Generate a continuous and differentiable function with the following properties:

Question

f(x) is decreasing at x = -6

f(x) has a local minimum at x = -2

f(x) has a local maximum at x = 2

I was able to get the minimum and maximum correct but my functions are always increasing at x = 6, any help would be appreciated!

Answer 1

f(x) = -x^3/3 +4x

take the derivative and set = 0 to solve for local max and min

f'(x) =-x^2+ 4 = 0

x = + or -2 are the local extrema, x=-2 is local min, =2 local max

f'(-6) =-36+4 = -32 < 0 which means downward sloping and f(x) decreasing at x=-6

you could add in a constant, any constant to f(x) such as f(x) =(-1/3)x^2 + 4x + 283, but c=0 works as well as any constant

you need a negative sign for the coefficient of the x^3 term, to make the 1st turning point a minimum, then the 2nd turning point is a maximum

with x=-2 and x=+2 as turning points

convert them to factors

(x+2)(x-2) =

x^2-4

integrate to get f(x)

f(x) = x^3/3 -4x, but then change the signs to get the 1st turning point as a local minimum

f(x) = -x^3/3 +4x