MCQ
Assertion $(A):$ If the function $f(x)=\frac{a e^x+b e^{-x}}{c e^x+d e^{-x}}$ is increasing function of $x$, then $b c > a d$.
Reason $(R):$ A function $f(x)$ is increasing if $f^{\prime}(x) > 0$ for all $x$.
  • A
    Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
  • Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
  • C
    $(A)$ is true but $(R)$ is false.
  • D
    $(A)$ is false but $(R)$ is true.

Answer

Correct option: B.
Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
$f^{\prime}(x)=\frac{2(a d-b c)}{\left(c e^x+d e^{-x}\right)^2}$and $f(x)$ is an increasing function.
$\therefore f^{\prime}(x) > 0$
$\Rightarrow \frac{2(a d-b c)}{\left(c e^x+d e^{-x}\right)^2} > 0$
$\Rightarrow 2(a d-b c) > 0$
$\Rightarrow a d > b c$
$ \Rightarrow b c < ad$

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