Let f - R - R be a differentiable function for all values of x and has a property that f x and f - x have opposite signs for all values of x- Then-A- f x is an increasing function-B- f x is a decreasing function-C- f 2 x is a decreasing function-D- - f x - is an increasing function-

Given that, fxf'x lt;0⇒12ddxf2x lt;0⇒ddxf2x lt;0Thus, nbsp;f2x nbsp;is nbsp;a nbsp;decreasing nbsp;function. ...

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Byju's Answer

Standard XII

Mathematics

Monotonically Decreasing Functions

Let f : R → R...

Question

Let $f : R \to R$ be a differentiable function for all values of x and has a property that f(x) and f '(x) have opposite signs for all values of x. Then,

f(x) is an increasing function.

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f(x) is a decreasing function.

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f^{2} (x)

is a decreasing function.

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|f(x)| is an increasing function.

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Solution

The correct option is C $f^{2} (x)$ is a decreasing function.
Given that,
$f (x) f' (x) < 0 \Rightarrow \frac{1}{2} \frac{d}{dx} (f^{2} (x)} < 0 \Rightarrow \frac{d}{dx} (f^{2} (x)} < 0 Thus, f^{2} (x) is a decreasing function .$

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f : R \to R

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f (x)

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f^{'} (x)

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Let f:R→R be a differentiable function for all values of x and has a property that f(x) and f '(x) have opposite signs for all values of x. Then,

The correct option is C f2(x) is a decreasing function.Given that, f(x)f'(x)<0⇒12ddx(f2(x)}<0⇒ddx(f2(x)}<0Thus, f2(x) is a decreasing function.

Let $f : R \to R$ be a differentiable function for all values of x and has a property that f(x) and f '(x) have opposite signs for all values of x. Then,

The correct option is C $f^{2} (x)$ is a decreasing function.
Given that,
$f (x) f' (x) < 0 \Rightarrow \frac{1}{2} \frac{d}{dx} (f^{2} (x)} < 0 \Rightarrow \frac{d}{dx} (f^{2} (x)} < 0 Thus, f^{2} (x) is a decreasing function .$