The local maximum value of the function f (x) = 3x⁴ + 4x³ - 12x² + 12 are at which point?

Concept:

Let f be a continuous function such that f '(p) = 0

• If f ''(p) > 0 then f has a local minimum at p.

• If f ''(p) < 0 then f has a local maximum at p.

Calculation:

f (x) = 3x4 + 4x³ - 12x2 + 12

⇒ f ' (x) = 12x³ + 12x² - 24x + 0      ----(1)

⇒ f ' (x) = 12x (x² + x - 2)

⇒ f ' (x) = 12x (x - 1)(x + 2)

Putting f ' (x) = 0

⇒ 12x (x - 1)(x + 2) = 0

⇒ x = 0, 1, -2 are the critical point

Finding f '' (x),

⇒ f '' (x) = 36x² + 24x - 24      [using (1)]

⇒ f '' (x) = 12 (3x² + 2x - 2)

Case 1: At x = 0,

f '' (x) = 12 (3(0)2 + 2(0) - 2)

⇒ f '' (x) = 12 (-2) = -24 < 0 

Since, f '' (x) < 0 at x = 0

∴ x = 0 is the point of local maxima

Thus, f(x) is maximum at x = 0.

Case 2: At x = 1

f '' (x) = 12 (3(1)2 + 2(1) - 2)

⇒ f '' (x) = 12 (3 + 2 - 2) = 36 > 0 

Since, f '' (x) > 0 at x = 1

∴ x = 1 is the point of local minima

Thus, f(x) is minimum at x = 1.

Case 3: At x = -2

f '' (x) = 12 (3(-2)2 + 2(-2) - 2)

⇒ f '' (x) = 12 (12 - 4 - 2) = 72 > 0 

Since, f '' (x) > 0 at x = -2

∴ x = -2 is the point of local minima

Thus, f(x) is minimum at x = -2.

Hence, at point x = 0, f(x) is maximum