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# Solution for The Function F ( X ) = Log E ( X 3 + √ X 6 + 1 ) is of the Following Types: (A) Even and Increasing (B) Odd and Increasing (C) Even and Decreasing (D) Odd and Decreasing - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

The function $f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)$ is of the following types:
(a) even and increasing
(b) odd and increasing
(c) even and decreasing
(d) odd and decreasing

#### Solution

(b) odd and increasing

$f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)$

$\Rightarrow f( - x) = \log_e \left( - x^3 + \sqrt{x^6 + 1} \right)$

$= \log_e \left\{ \frac{\left( - x^3 + \sqrt{x^6 + 1} \right)\left( x^3 + \sqrt{x^6 + 1} \right)}{x^3 + \sqrt{x^6 + 1}} \right\}$

$= \log_e \left( \frac{x^6 + 1 - x^6}{x^3 + \sqrt{x^6 + 1}} \right)$

$= \log_e \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right)$

$= - \log_e \left( x^3 + \sqrt{x^6 + 1} \right)$

$= - f(x)$

$\text { Hence,} f( - x) = - f(x)$

$\text { Therefore, it is an odd function } .$

$f(x) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)$

$\frac{d}{dx}\left\{ f(x) \right\} = \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right) \times \left( 3 x^2 + \frac{1}{2\sqrt{x^6 + 1}} \times 6 x^5 \right)$

$= \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right) \times \left( \frac{6 x^2 \sqrt{x^6 + 1} + 6 x^5}{2\sqrt{x^6 + 1}} \right)$

$= \left( \frac{1}{x^3 + \sqrt{x^6 + 1}} \right) \times \left\{ \frac{6 x^2 \left( \sqrt{x^6 + 1} + x^3 \right)}{2\sqrt{x^6 + 1}} \right\}$

$= \left( \frac{6 x^2}{2\sqrt{x^6 + 1}} \right) > 0$

$\text { Therefore, given function is an increasing function } .$

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#### Video TutorialsVIEW ALL [3]

Solution The Function F ( X ) = Log E ( X 3 + √ X 6 + 1 ) is of the Following Types: (A) Even and Increasing (B) Odd and Increasing (C) Even and Decreasing (D) Odd and Decreasing Concept: Increasing and Decreasing Functions.
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