SATHEE: Differential Equations 1 Question 20

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

(b) $f^{'} (x) > f (x), 0 < x < \frac{1}{4}$

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

Show Answer

Solution:

Let $φ (x) = e^{- x} f (x)$

Here,

$φ^{'} (x) < 0, x \in 0, \frac{1}{4}$

and

$φ^{'} (x) > 0, x \in \frac{1}{4}, 1$

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

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

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Differential Equations 1 Question 20

20. If the function e−xf(x) assumes its minimum in the interval [0,1] at x=1/4, then which of the following is true?

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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?