Explanation:
f(x) = 7 + 3x − x³
Find the derivative:
f'(x) = 3 − 3x²
Set to 0 and solve for x:
0 = 3 − 3x²
0 = 1 − x²
x² = 1
x = ±1
Plug into f(x):
f(-1) = 5
f(1) = 9
We need to determine if these are minima or maxima.
Find and evaluate the second derivative:
f"(x) = -6x
f"(-1) = 6 > 0 (positive or concave up)
f"(1) = -6 < 0 (negative or concave down)
f(-1) = 5 is a local minimum.
f(1) = 9 is a local maximum.