Using the definition

And plugging in our function, we get that

.

if we factor out inside the limit we get

since the term doesn't contain an h we can factor it out, and then divide by both sides, getting that

but we know that

So we find that the limit is equal to .

To find the absolute extrema of a function on a closed interval, one must first take the first derivative of the function.

The derviatve of this function by the power rule is as follows:

The relative extrema is when the first derivative is equal to 0, that is, there is a change in slope.

Solving for x when it is equal to zero derives:

Diving by 6 and factoring gives or however, since we are concerned with the interval (-2,0) our x value is -1.

We now however must find the value of f(x) at -1

To find the absolute extrema of a function on a closed interval, one must first take the first derivative of the function.

The derviatve of this function by the power rule is as follows:

The relative extrema is when the first derivative is equal to 0, that is, there is a change in slope.

Solving for x when it is equal to zero derives:

Diving by 6 and factoring gives or however, since we are concerned with the interval (-2,0) our x value is -1.

We now however must find the value of f(x) at -1

To evaluate this derivative, we use the Product Rule.

. Use the Product Rule. Keep in mind that the derivative of involves the Chain Rule.

. Factor out an .

To evaluate this derivative, we use the Product Rule.

. Use the Product Rule. Keep in mind that the derivative of involves the Chain Rule.

. Factor out an .

We use the product rule to differentiate this function. Applying it looks like this:

This simplifies to:

We apply the chain rule to differentiate , which becomes . Plugging this into the above equation gives us:

or

We use the product rule to differentiate this function. Applying it looks like this:

This simplifies to:

We apply the chain rule to differentiate , which becomes . Plugging this into the above equation gives us:

or

For the term, we simply use the power rule to abtain . Since is a constant (not a variable), we treat it as such. The derivative of any constant (or "stand-alone number") is .

For the term, we simply use the power rule to abtain . Since is a constant (not a variable), we treat it as such. The derivative of any constant (or "stand-alone number") is .