# How do you find the exact relative maximum and minimum of the polynomial function of #2x^3 -23x^2+78x-72#?

There is a minimum at

There is a maximum at

Given -

#y=2x^3-23x^2+78x-72#

#dy/dx=6x^2-46x+78#

#(d^2y)/(dx^2)=12x-46#

#dy/dx=0 => 6x^2-46x+78=0#

#x=(-b+-sqrt(b^2 -4ac))/(2xxa)#

#x=(-(-46)+-sqrt((-46^2) -(4xx6xx78)))/(2xx6)#

#x=(46+-sqrt(2116-1872))/(2xx6)#

#x=(46+-sqrt(244))/(12)#

#x=(46+-15.62)/(12)#

#x=(46+15.62)/(12)=61.62/12=5.135#

#x=(46-15.62)/(12)=30.38/12=2.531#

At

#(d^2y)/(dx^2)=12(5.135)-46=61.62-46=15.62 >0#

There is a minimum at

At

#(d^2y)/(dx^2)=12(2.531)-46=30.37-46=-15.628<0#

There is a maximum at

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To find the exact relative maximum and minimum of the polynomial function (2x^3 - 23x^2 + 78x - 72), you need to follow these steps:

- Find the derivative of the function.
- Set the derivative equal to zero and solve for (x). These values of (x) represent critical points.
- Use the second derivative test or the first derivative test to determine whether each critical point corresponds to a relative maximum, minimum, or neither.

Let's find the derivative of the function: (f'(x) = 6x^2 - 46x + 78)

Now, set (f'(x) = 0) and solve for (x): (6x^2 - 46x + 78 = 0)

Using the quadratic formula, you find the roots of this equation, which are the critical points.

Next, you can use either the second derivative test or the first derivative test to determine the nature of these critical points and thus find the relative maximum and minimum of the function.

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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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