PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for Find the Interval in Which the Following Function Are Increasing Or Decreasing F ( X ) = Log ( 2 + X ) − 2 X 2 + X , X ∈ R ? - PUC Karnataka Science Class 12 - Mathematics

#### Question

Find the interval in which the following function are increasing or decreasing $f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R$ ?

#### Solution

$\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.$

$\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .$

$f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R$

$f'\left( x \right) = \frac{1}{\left( 2 + x \right)} - \frac{\left[ \left( 2 + x \right)2 - 2x \right]}{\left( 2 + x \right)^2}$

$= \frac{\left( 2 + x \right) - \left[ 4 + 2x - 2x \right]}{\left( 2 + x \right)^2}$

$= \frac{2 + x - 4}{\left( 2 + x \right)^2}$

$= \frac{\left( x - 2 \right)}{\left( 2 + x \right)^2}, x \neq - 2$

$\text{ Here, x = 2 is the critical point}.$

$\text { The possible intervals are }\left( - \infty , 2 \right)\text { and }\left( 2, \infty \right). .....(1)$

$\text { For f(x) to be increasing, we must have }$

$f'\left( x \right) > 0$

$\Rightarrow \frac{\left( x - 2 \right)}{\left( 2 + x \right)^2} > 0$

$\Rightarrow x - 2 > 0, x \neq - 2$

$\Rightarrow x > 2$

$\Rightarrow x \in \left( 2, \infty \right) \left[ \text { From eq. } (1) \right]$

$\text{ So,f(x)is increasing on x }\in \left( 2, \infty \right) .$

$\text { Forf(x) to be decreasing, we must have }$

$f'\left( x \right) < 0$

$\Rightarrow \frac{\left( x - 2 \right)}{\left( 2 + x \right)^2} < 0$

$\Rightarrow x - 2 < 0, x \neq - 2$

$\Rightarrow x < 2$

$\Rightarrow x \in \left( - \infty , 2 \right) \left[ \text { From eq.} (1) \right]$

$\text { So,f(x)is decreasing on x }\in \left( - \infty , 2 \right) .$

Is there an error in this question or solution?

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Solution Find the Interval in Which the Following Function Are Increasing Or Decreasing F ( X ) = Log ( 2 + X ) − 2 X 2 + X , X ∈ R ? Concept: Increasing and Decreasing Functions.
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