(B) Given the function $f(x) = e^{ax} + e^{-ax}$.Find the derivative with respect to $x$:$f'(x) = a e^{ax} - a e^{-ax} = a(e^{ax} - e^{-ax})$.For the

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(B) Given the function $f(x) = e^{ax} + e^{-ax}$.Find the derivative with respect to $x$:$f'(x) = a e^{ax} - a e^{-ax} = a(e^{ax} - e^{-ax})$.For the function to be monotonically decreasing,we require $f'(x) Since $a 0$.This inequality holds when $e^{ax} > e^{-ax}$.Taking the natural logarithm on both sides:$ax > -ax$$2ax > 0$.Since $a $x Therefore,the function is monotonically decreasing for $x < 0$.

Class 12 Mathematics Applications of Derivatives Increasing and Decreasing function

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If $a < 0$,then the function $f(x) = e^{ax} + e^{-ax}$ is monotonically decreasing for all values of $x$ where ...

A
$x > 0$
B
$x < 0$
C
$x > 1$
D
$x < 1$

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Class 12

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Applications of Derivatives

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Increasing and Decreasing function

The function $f(x) = x^3 - 3x^2 - 24x + 5$ is an increasing function in the interval given below:

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If $f(x) = x e^{x(1-x)}, x \in R$,then $f(x)$ is

MediumAP EAMCET 2025

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The interval in which the function $f(x) = x^{3} - 6x^{2} + 9x + 10$ is increasing is:

EasyKCET 2019

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Which of the following functions is decreasing in the interval $\left(0, \frac{\pi}{8}\right)$?

EasyGUJCET 2023

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The function $f(x) = \sin^4 x + \cos^4 x$ increases if

MediumMHT CET 2025

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If $a < 0$,then the function $f(x) = e^{ax} + e^{-ax}$ is monotonically decreasing for all values of $x$ where ...

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