Final answer:
To determine whether the function is concave up or concave down at the indicated points, we need to find the second derivative of the function. By evaluating the second derivative at x = -1 and x = 6, we can determine the concavity at these points.
Step-by-step explanation:
To determine whether the function is concave up or concave down at the indicated points, we need to find the second derivative of the function. Let's start by finding the first derivative.
f(x) = x^3 - 3x^2 + 6
f'(x) = 3x^2 - 6x
Now, let's find the second derivative.
f''(x) = 6x - 6
To determine the concavity, we need to evaluate the second derivative at the given points.
a. x = -1:
f''(-1) = 6(-1) - 6 = -12
Since the second derivative is negative, the function is concave down at x = -1.
b. x = 6:
f''(6) = 6(6) - 6 = 30
Since the second derivative is positive, the function is concave up at x = 6.