Question

Which of the following is true for $0<x<1$?

• A. $-\frac{1}{4}<f\left(x\right)<1$
• B. $0<f\left(x\right)<\infty$
• C. $-\infty <f\left(x\right)<0$
• D. $-\frac{1}{2}<f\left(x\right)<\frac{1}{2}$

Answer 1

The correct option is C $-\infty <f\left(x\right)<0$

$\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y\ge e^x$

$\Rightarrow e^{-x}\frac{d^{2}y}{d{x}^2}-2e^{-x}\frac{dy}{dx}+e^{-x}y\ge 1$

$\Rightarrow \frac{d^2}{d{x}^2}\left(y{e}^{-x}\right)\ge 1$

$\Rightarrow y{e}^{-x}$ is concave up

Hence, $-\infty <f\left(x\right)<0$