Question

Determine whether the function is concave up or concave down at the indicated points:

f(x)=x3−3x2+6f(x) = x³ - 3x² + 6f(x)=x3−3x2+6
a) x=−1x = -1x=−1 concave up concave down
b) x=6x = 6x=6 concave up concave down

Answer 1

To determine whether the function is concave up or concave down at the given points, find the second derivative of the function and evaluate it at those points. The second derivative will indicate the concavity of the function.

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.
f'(x) = 3x^2 - 6x
Now, let's find the second derivative by taking the derivative of the first derivative:
f''(x) = 6x - 6

To determine concavity, we need to evaluate the second derivative at the given points:

a) For x = -1: f''(-1) = 6(-1) - 6 = -12
Since the second derivative is negative, the function is concave down at x = -1.

b) For x = 6: f''(6) = 6(6) - 6 = 30
Since the second derivative is positive, the function is concave up at x = 6.