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

Answer 1

See below.

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#.

Create a graph using the graphing tool graph{y=x^3+6x^2-36 [-80, 80, -40, 40]} to confirm.

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Answer 2

To find the exact relative maximum and minimum of the polynomial function g(x) = x^3 + 6x^2 - 36, you first find its critical points by finding where its derivative equals zero. Then, you determine the nature of these critical points by analyzing the sign of the derivative around each point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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