To find the maximum, we must find where the graph shifts from increasing to decreasing. To find out the rate at which the graph shifts from increasing to decreasing, we look at the second derivative and see when the value changes from positive to negative.

That is to say, we will look at the second derivative and see where (if at all) the graph crosses the x-axis and is moving from a positive y-value to a negative y-value.

Now we must find the second derivative. Unfortunately, the derivatives of trig functions must be memorized. The first derivative is:

First find the derivative, which is:

Set that equal to zero to get your critical points:

You also have to take into account -1 and 2 as critical points since those are your endpoints. Evaluate each of these critical points in the original function:

The greatest is your maximum. The x coordinate is 2.

To find the maximum, we need to find the critical points. To do that, we need to take the derivatie of the function.

And we can see that  and  are critical points for this function.

An easy way to see which is the maximum and which is the minimum is to plug in the values of the critical points into the original equation.

and

Therefore, the maximum value is 3, and the value at which the function reaches the maximum is 0.

To find potential minima of the function, take the first derivative of  using the power rule.

Set the derivative to 0:

We solve for  to obtain  and then plug in 0.5 into the original function to obtain the answer of

We can double check that  is indeed a minimum by using the second derivative test

which means the function is concave up, so that the point we found is a minimum.

First, find the derivative, which is:

Set that equal to 0 to find your critical points:

Evaluate f at each critical point and endpoint over [-2, 2]:

The least of those numbers is the minimum. Because f(-2)=-32, that is the minimum.

To find the maximum, we need to look at the first derivative. 

To find the first derivative, we can use the power rule. To do that, we lower the exponent on the variables by one and multiply by the original exponent.

We're going to treat  as  since anything to the zero power is one.

Notice that  since anything times zero is zero.

When looking at the first derivative, remember that if the output of this equation is positive, the original function is increasing. If the derivative is negative, then the function is decreasing.

Because we want the MINIMUM, we want to see where the derivative changes from negative to positive.

Notice that  has a root when . In fact, it changes from negative to positive at that particular point. This is the local minimum in the interval .

Integrate

1) Apply the sum rule for integration, 

2) Integrate each individual term and include a constant of integration,  

Further Discussion

Since indefinite integration is essentially a reverse process of differentiation, check your result by computing its' derivative. 

This is the same function we integrated, which confirms our result. Also, because the derivative of a constant is always zero, we must include "C" in our result since any constant added to any function will produce the same derivative.