Show that f(x) = (x – 1)ex + 1 is an increasing function for all x

1.	Show that f(x) = (x – 1)ex + 1 is an increasing function for all x > 0.
Answer» Given as f(x) = (x – 1)e^x + 1 Differentiate the given equation with respect to x, we get ⇒ f'(x) = (d/dx)((x - 1)e^x + 1) ⇒ f’(x) = e^x + (x – 1) e^x ⇒ f’(x) = e^x(1+ x – 1) ⇒ f’(x) = x e^x As given x > 0 ⇒ e^x > 0 ⇒ x e^x > 0 ⇒ f’(x) > 0 Thus, condition for f(x) to be increasing Hence f(x) is increasing on interval x > 0

Show that f(x) = (x – 1)ex + 1 is an increasing function for all x > 0.

Answer»

Given as f(x) = (x – 1)e^x + 1

Differentiate the given equation with respect to x, we get

⇒ f'(x) = (d/dx)((x - 1)e^x + 1)

⇒ f’(x) = e^x + (x – 1) e^x

⇒ f’(x) = e^x(1+ x – 1)

⇒ f’(x) = x e^x

As given x > 0

⇒ e^x > 0

⇒ x e^x > 0

⇒ f’(x) > 0

Thus, condition for f(x) to be increasing

Hence f(x) is increasing on interval x > 0