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}; S₂ = {0, 1} 

• S₁ = {-2, 0}; S₂ = {1} 

• S₁ = {-2, 1}; S₂ = {0} 

• 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 S₁ = {-2, 1}; S₂ = {0}.

Explanation:

f(x) = 9x⁴ + 12x³ - 36x² + 25

∴ f'(x) = 36x³ + 36x² - 72x
          = 36x (x - 1) (x + 2)
f'(-, +)             (+, -)    (-, +)  
   -2                   0               1

f' changes sign from -ve to +ve in the neighbourhood of x = -2 and x = 1.

∴ S₁ = {-2, 1}

f' changes sign from +ve to -ve in the neighbourhood of x = 0.

∴ S₂ = {0}