Practicing Success
If $a < 0, f(x)=e^{a x}+e^{-a x}$ and S = {x : f(x) is monotonically increasing}, then S equals |
{x : x > 0} {x : x < 0} {x : x > 1} {x : x < 1} |
{x : x > 0} |
We have, $f(x)=e^{a x}+e^{-a x} \Rightarrow f'(x)=a\left(e^{a x}-e^{-a x}\right)$ For f(x) to be increasing, we must have $f'(x)>0$ $\Rightarrow a\left(e^{a x}-e^{-a x}\right)>0$ $\Rightarrow e^{a x}-e^{-a x}<0$ [∵ a < 0] $\Rightarrow e^{-b x}-e^{b x}<0$ where, a = -b and b > 0 $\Rightarrow e^{b x}-e^{-b x}>0 \Rightarrow x > 0$ Hence, S = {x : x > 0} |