f(x)=x^4-16x^2. Is this function concave up or down at x=-2?
I will be happy to critique your thinking.
To determine if the function is concave up or down at a specific point, you need to find the second derivative of the function and evaluate it at that point.
First, let's find the first derivative of f(x):
f'(x) = 4x^3 - 32x
Now, let's find the second derivative by differentiating f'(x) with respect to x:
f''(x) = 12x^2 - 32
To determine whether the function is concave up or down at x = -2, substitute x = -2 into the second derivative:
f''(-2) = 12(-2)^2 - 32
= 12(4) - 32
= 48 - 32
= 16
If the second derivative is positive, then the function is concave up at that point. If it is negative, then the function is concave down.
In this case, f''(-2) = 16, which is positive. Therefore, the function f(x) = x^4 - 16x^2 is concave up at x = -2.
Remember, to determine the concavity at a specific point, you need to find the second derivative and evaluate it at that point.