How do you find the exact relative maximum and minimum of the polynomial function of g(x) =x^3 + 6x^2 – 36?

1 Answer
Mar 9, 2016

See below.

Explanation:

Given
g(x)=x^3+6x^2-36

To find the maximum and minimum of the function find first derivative and set it equal to zero.
g'(x)=3x^2+12x=0

After factoring out 3x we obtain
3x(x+4)=0

Since 3!=0, therefore
x=0, and x=-4 are the two points of inflection.

For x=0, we obtain y=-36, and for x=-4, we get y=-4

Hence (0,-36) and (-4,-4) are the points of interest.

Now to ascertain the second derivative of the given function
g''(x)=6x+12

At the first point (0,-36)
g''(0)=6xx0+12=12, a positive quantity. It is a local minimum for the value of x.

Similarly at the point (-4,-4),
g''(-4)=6xx(-4)+12=-12 a negative quantity. It is a local maximum for the value of x.

Verify by drawing a graph with the graphing tool
graph{y=x^3+6x^2-36 [-80, 80, -40, 40]}