Show that f(x) = x3 – 15x2 + 75x – 50 is an increasing function

1.	Show that f(x) = x3 – 15x2 + 75x – 50 is an increasing function for all x ϵ R.
Answer» Given as f(x) = x³ – 15x² + 75x – 50 f'(x) = (d/dx)(x³ – 15x² + 75x – 50) ⇒ f’(x) = 3x² – 30x + 75 ⇒ f’(x) = 3(x² – 10x + 25) ⇒ f’(x) = 3(x – 5)² As given x ϵ R ⇒ (x – 5)² > 0 ⇒ 3(x – 5)² > 0 ⇒ f’(x) > 0 Thus, condition for f(x) to be increasing Hence, f(x) is increasing on interval x ∈ R

Show that f(x) = x3 – 15x2 + 75x – 50 is an increasing function for all x ϵ R.

Answer»

Given as f(x) = x³ – 15x² + 75x – 50

f'(x) = (d/dx)(x³ – 15x² + 75x – 50)

⇒ f’(x) = 3x² – 30x + 75

⇒ f’(x) = 3(x² – 10x + 25)

⇒ f’(x) = 3(x – 5)²

As given x ϵ R

⇒ (x – 5)² > 0

⇒ 3(x – 5)² > 0

⇒ f’(x) > 0

Thus, condition for f(x) to be increasing

Hence, f(x) is increasing on interval x ∈ R