Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x

1.	Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ϵ R.
Answer» Given as f(x) = x⁹ + 4x⁷ + 11 Differentiate the above equation with respect to x, we get ⇒ f'(x) = (d/dx)(x⁹ + 4x⁷ + 11) ⇒ f’(x) = 9x⁸ + 28x⁶ ⇒ f’(x) = x⁶(9x² + 28) Also, as given x ϵ R ⇒ x⁶ > 0 and 9x² + 28 > 0 ⇒ x⁶(9x² + 28) > 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) = x9 + 4x7 + 11 is an increasing function for all x ϵ R.

Answer»

Given as f(x) = x⁹ + 4x⁷ + 11

Differentiate the above equation with respect to x, we get

⇒ f'(x) = (d/dx)(x⁹ + 4x⁷ + 11)

⇒ f’(x) = 9x⁸ + 28x⁶

⇒ f’(x) = x⁶(9x² + 28)

Also, as given x ϵ R

⇒ x⁶ > 0 and 9x² + 28 > 0

⇒ x⁶(9x² + 28) > 0

⇒ f’(x) > 0

Thus, condition for f(x) to be increasing

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