To find the critical points, we first take the first derivative using the Power Rule.

.

Therefore,

. 

From here we need to set the derivative equal to zero and solve to x.

The critical point occurs when . Now to get the y value of the point we will need to plug in x=0 into the original equation.

. This gives us a critical point at .

To find if it is a max or min, we take the second derivative

Since the second derivative is negative, it is a maximum.

Determine the derivative of this function.

Set the derivative function equal to zero and solve for .

Since the parabola has a positive coefficient for , the parabola will open upwards, and therefore will have a minimum.

Substitute back into the original function and find the y-value of the minimum.

The parabola has a minimum at the point .

Let's find the first derivative to locate the relative maxima and minima.

Now we set it equal to zero to find the x values of these critical points.

So the equation is 0 where x is -2, 0, or 5. Now let's find the second derivative so that we know which of these locations are maxima and which are minima.

So the function has a relative maximum at x=-5.

So the function has a relative minimum at x=0.

So the function has a relative maximum at x=2.

Thus there is only one relative minimum in this function, and it occurs at x=0. We need to plug this into the original function to find the y-coordinate of the point.

So our point is (0,8).

We need to find out where the first derivative is equal to zero to find the locations of the possible maxima and minima.

 or .

We see that the function is a quadratic, so its graph will be a parabola. Thus, we know it will have only one relative max or min. Since the leading coefficient of  is a positive 6, we also know that the parabola opens upward. Hence, the relative extremum is a minimum.

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Taking the second derivative:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Finally, we take the second derivative:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Then, taking the second derivative of the function:

We first apply Power Rule.

First Derivative :

So result is

Anything to a power of  is 

First Derivative is

Second Derivative :

Any derivative of a constant is 

Second Derivative of  with respect to  is