(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 Questions

Class 12

Mathematics Questions

Chapter

Applications of Derivatives

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Increasing and Decreasing function

In which of the following intervals is the function $f(x) = 2x^3 - 15x^2 + 36x + 1$ strictly decreasing?

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Function $f(x) = \frac{\lambda \sin x + 3 \cos x}{2 \sin x + 6 \cos x}$ is monotonic increasing,then :

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The interval of the decreasing function $f(x) = x^3 - x^2 - x - 4$ is

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If $y = ax^3 + 3x^2 + (2a + 1)x + 1000$ is a strictly increasing function for all values of $x$,then:

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Prove that the function given by $f(x) = \cos x$ is neither increasing nor decreasing in $(0, 2\pi)$.

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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 ...

Similar Questions

In which of the following intervals is the function $f(x) = 2x^3 - 15x^2 + 36x + 1$ strictly decreasing?

Function $f(x) = \frac{\lambda \sin x + 3 \cos x}{2 \sin x + 6 \cos x}$ is monotonic increasing,then :

The interval of the decreasing function $f(x) = x^3 - x^2 - x - 4$ is

If $y = ax^3 + 3x^2 + (2a + 1)x + 1000$ is a strictly increasing function for all values of $x$,then:

Prove that the function given by $f(x) = \cos x$ is neither increasing nor decreasing in $(0, 2\pi)$.

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