Question

The local maximum value of the function f(x) = 3x⁴ + 4x³ - 12x² + 12 is at x = :
1. 1
2. 2
3. -2
4. 0
5. None of these

Answer 1

Correct Answer - Option 4 : 0

Concept:

For a function y = f(x):

• A relative (local) maxima is a point where the function f(x) changes its direction from increasing to decreasing.
• A relative (local) minima is a point where the function f(x) changes its direction from decreasing to increasing.
• At the point of relative (local) maxima or minima, f'(x) = 0.
• At the point of relative (local) maxima, f''(x) < 0.
• At the point of relative (local) minima, f''(x) > 0.

 

Calculation:

For the given function f(x) = 3x4 + 4x3 - 12x2 + 12, first let's find the points of local maxima or minima:

f'(x) = 12x³ + 12x² - 24x = 0

⇒ 12x(x² + x - 2) = 0

⇒ x(x + 2)(x - 1) = 0

⇒ x = 0 OR x = -2 OR x = 1.

f''(x) = 36x² + 24x - 24.

f''(0) = 36(0)2 + 24(0) - 24 = -24.

f''(-2) = 36(-2)2 + 24(-2) - 24 = 144 - 48 - 24 = 72.

f''(1) = 36(1)2 + 24(1) - 24 = 36 + 24 - 24 = 36.

Since, at x = 0 the value f''(0) = -24 < 0, the local maximum value of the function occurs at x = 0.