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Home » Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.
Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.
Application of Derivatives | Class 12 | Exercise 1 | Maths | NCERT Exemplar
Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.

Given,

 

    \[f\text{ }\left( x \right)\text{ }=\text{ }2x\text{ }+\text{ }bed\text{ }1\text{ }x\text{ }+\text{ }log\text{ }\left[ \surd \left( 1\text{ }+\text{ }x2 \right)\text{ }\text{ }x \right]\]

differentiating the two sides w.r.t. x, we get

NCERT Exemplar Solutions Class 12 Mathematics Chapter 6 - 28

On figuring out both the sides, we get

 

    \[4x4\text{ }+\text{ }1\text{ }+\text{ }4x2\text{ }\ge \text{ }1\text{ }+\text{ }x2\]

    \[4x4\text{ }+\text{ }4x2\text{ }\text{ }x2\text{ }\ge \text{ }0\]

    \[4x4\text{ }+\text{ }3x2\text{ }\ge \text{ }0\]

    \[x2\left( 4x2\text{ }+\text{ }3 \right)\text{ }\ge \text{ }0\]

The above is valid for any worth of x ∈ R.

 

In this manner, the given capacity is an expanding capacity over R.

i

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