properties of real differentiable function

Question

problem

I have tried to investigate f according to given criteria but dont seem to go nowhere,

supposedly i have could show that according to given criteria f must be either strictly positive and growing or strictly negative and decreasing

but i do not know how to go further in proving the given statement

thanks in advance for your help

Answer 1

Note that $$ \left(f(x)e^{-cx}\right)'=f'(x)e^{-cx}-cf(x)e^{-cx}\geq 0. $$ Which implies that $$ f(x)\geq e^{cx} \frac{f(x_0)}{e^{cx_0}}=f(x_0) e^{c(x-x_0)} $$ for all $x> x_0$. Applying continuity at $a$ gives you the desired. You can prove the other inequality similarly.