Question

Find the maximum and minimum values of the function: x^3 - 3x^2 - 9x + 12

Answer 1

Answer:

maximum and minimum values are 17 and (-15) respectively.

Step-by-step explanation:

let the given function be , f(x) = x³ - 3x² -9x +12

then, differentiating f(x) we get ,

f₁(x) = 3x² - 6x -9

again differentiating f₁(x) we get ,

f₂(x) = 6x -6

then, f₁(x) = 0

⇒3x² - 6x -9 = 0

⇒ x² - 2x -3 = 0

⇒ x² + ( -3 +1 )x +(-3)×1 =0

⇒(x-3)(x+1) =0

⇒x = 3 , (-1)

then , f₂(3) = 6×3 - 6 = 12 > 0  ;  minima of f(x) lies at x = 3

and , f₂(-1) = 6×(-1) -6 = -12 < 0  ; maxima of f(x) lies at x = (-1)

∴ maximum value of f(x) is , f(-1) = (-1)³ - 3(-1)² - 9(-1) +12= 17

∴ minimum value of f(x) is , f(3) = (3)³ - 3(3)² - 9(3) +12 = (-15)