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) $f^{\prime}(x)<f(x), \frac{3}{4}<x<1$

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Solution:

  1. Let $\varphi(x)=e^{-x} f(x)$

Here,

$$ \varphi^{\prime}(x)<0, x \in 0, \frac{1}{4} $$

and

$$ \varphi^{\prime}(x)>0, x \in \frac{1}{4}, 1 $$

$\Rightarrow e^{-x} f^{\prime}(x)-e^{-x} f(x)<0, x \in 0, \frac{1}{4}$

$\Rightarrow f^{\prime}(x)<f(x), 0<x<\frac{1}{4}$



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