Differential Equations 1 Question 21
21. Which of the following is true?
(a) $0<f(x)<\infty$
(b) $-\frac{1}{2}<f(x)<\frac{1}{2}$
(c) $-\frac{1}{4}<f(x)<1$
(d) $-\infty<f(x)<0$
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Solution:
- Here, $f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}$
$\Rightarrow f^{\prime \prime}(x) e^{-x}-f^{\prime}(x) e^{-x}-f^{\prime}(x) e^{-x}+f(x) e^{-x} \geq 0$
$\Rightarrow \frac{d}{d x}{f^{\prime}(x) e^{-x} }-\frac{d}{d x}{f(x) e^{-x} } \geq 1$
$\Rightarrow \quad \frac{d}{d x}{f^{\prime}(x) e^{-x}-f(x) e^{-x} } \geq 1$
$\Rightarrow \frac{d^{2}}{d x^{2}}{e^{-x} f(x) } \geq 1, \forall x \in[0,1]$
$\therefore \varphi(x)=e^{-x} f(x)$ is concave function.
$$ \begin{array}{lc} & f(0)=f(1)=0 \\ \Rightarrow & \varphi(0)=0=f(1) \\ \Rightarrow & \varphi(x)<0 \\ \Rightarrow & e^{-x} f(x)<0 \\ \therefore & f(x)<0 \end{array} $$
