Question

If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x⁴ + 12x³ – 36x² + 25, x ∈ R, then ______.

Options

• S₁ = {–2, 1}; S₂ = {0}

• S₁ = {–2, 0}; S₂ = {1}

• S₁ = {–2}; S₂ = {0, 1}

• S₁ = {–1}; S₂ = {0, 2}

Answer 1

If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x⁴ + 12x³ – 36x² + 25, x ∈ R, then `underlinebb(S_1 = {-2, 1}; S_2 = {0})`.

Explanation:

f(x) = 9x⁴ + 12x³ – 36x² + 25

f'(x) = 36x³ + 36x² - 72x

= 36x(x² + x - 2)

= 36x(x + 2)(x - 2)

f'(x) = 0  for maximum and minimum

36x(x + 2)(x - 1) = 0

x = 0, 1, -2

f''(x) = 108x² + 72x – 72

= 36(3x² + 2x – 2)

 f''(0) = –2 then maximum

if f''(c) is  –ve then maximum

f'(1) = +ve  Minimum at  x = 1

f''(0) is +ve  then minimum

f''(-2) = +ve so Minimum