Question

Find the intervals Where f(x)=x⁴+32x is increasing or decreasing. x ∈R.

Answer 1

concept : any function , y = f(x) is increasing on (a, b) only if f'(x) > 0 on (a, b) while decreasing on (a, b) only if f'(x) < 0 on (a, b).

given, function, f(x) = x⁴ + 32x

differentiating f(x) with respect to x,

f'(x) = 4x³ + 32

now, f'(x) = 4x³ + 32 = 0

⇒4(x³ + 8) = 0

⇒x³ = -8 = (-2)³

⇒x = -2

case 1 : when x > -2

f'(x) = 4(x³ + 8) > 0

hence, function f(x) = x⁴ + 32x is increasing on x ∈ (-2, ∞).

case 2 : when x < -2

f'(x) = 4(x³ + 8) < 0

hence, function is decreasing on x ∈ (-∞, -2)