#### Question

Show that f(x) = (x − 1) e^{x} + 1 is an increasing function for all x > 0 ?

#### Solution

\[f\left( x \right) = \left( x - 1 \right) e^x + 1\]

\[f'\left( x \right) = \left( x - 1 \right) e^x + e^x \]

\[ = x e^x - e^x + e^x \]

\[ = x e^x \]

\[\text { Given }:x > 0 \]

\[\text { We know,}\]

\[ e^x > 0\]

\[\Rightarrow x e^x > 0\]

\[ \Rightarrow f'\left( x \right) > 0, \forall x > 0\]

\[\text { So },f(x)\text { is increasing on for all }x>0.\]

Is there an error in this question or solution?

Solution Show that F(X) = (X − 1) Ex + 1 is an Increasing Function for All X > 0 ? Concept: Increasing and Decreasing Functions.