Differential Equations 1 Question 20
20. If the function $e^{-x} f(x)$ assumes its minimum in the interval $[0,1]$ at $x=1 / 4$, then which of the following is true?
(a) $f^{\prime}(x)<f(x), \frac{1}{4}<x<\frac{3}{4}$
(b) $f^{\prime}(x)>f(x), 0<x<\frac{1}{4}$
(c) $f^{\prime}(x)<f(x), 0<x<\frac{1}{4}$
(d) $f^{\prime}(x)<f(x), \frac{3}{4}<x<1$
Show Answer
Solution:
- Let $\varphi(x)=e^{-x} f(x)$
Here,
$$ \varphi^{\prime}(x)<0, x \in 0, \frac{1}{4} $$
and
$$ \varphi^{\prime}(x)>0, x \in \frac{1}{4}, 1 $$
$\Rightarrow e^{-x} f^{\prime}(x)-e^{-x} f(x)<0, x \in 0, \frac{1}{4}$
$\Rightarrow f^{\prime}(x)<f(x), 0<x<\frac{1}{4}$
