For f(x)=2(x+5)^3 +7

Question

Find and classify the extreme values, determine where the function is increasing and decreasing, where it is concave up and concave down, and any points of inflection.

I understand this is using the first and second derivative tests, but I can't find any zeroes for the derivative of this function. Am I doing something wrong?

Answer 1

f(x)=0=2(x+5)^3 +7
(x+5)^3=-7/2
x+5= -cubroot(7/2)
x=-5-cubroot(7/2)

Those are the zeroes, but I don't see in the problem wording it asked for a zero

Answer 2

y = 2 (x+5)^3 + 7

to make this really easy you can say
z = x+5
then dz/dx = 1

y = 2 z^3 + 7
dy/dz = 6 z^2
d^2y/dz^2 = 12 z

so
dy/dx = dy/dz * dz/dx = 6(x^2+10x+25)
when is that zero ?
(x+5)(x+5) = 0
x = -5 twice

what is d/dx (dy/dx)?
6 ( 2 x + 10)
when x = -5
2(-5)+10 = 0
so that is not a max or a min but an inflection point.

Answer 3

Think of the graph for y = x^3
It has no max/min, but just an inflection point at x=0.

You function is just the same, only stretched by 2, shifted left by 5, and shifted up by 7.