# How do you find local maximum value of f using the first and second derivative tests: #f(x)= x^2 + 8x -12#?

##### 1 Answer

#### Explanation:

Find

To determine whether or not the point is a local maximum, you could use either the first or second derivative tests.

For the **first derivative test** , create a sign chart where the important values are the critical numbers. If the signs change from positive to negative, the point is a local maximum.

For the **second derivative test** , plug the critical number(s) into the second derivative. If the value is negative, the function is concave down at that point meaning the point is a local maximum.

Now, let's do the work:

**First derivative test:**

Since the sign of the first derivative goes from negative to positive, there is a local *minimum* when

We can prove the same thing with the...

**Second derivative test:**

Thus, the second derivative is ALWAYS positive, and the function is always concave up, which results in a local *minimum.*

Therefore, the function has no local maxima.

We can check a graph, even though it is obvious that the graph will form a parabola facing up, which will have only a minimum at its vertex:

graph{x^2+8x-12 [-65, 66.67, -36.6, 29.23]}