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]}