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Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions1 Mark
Question
The function $\text{f}(\text{x})=2\log(\text{x}-2)-\text{x}^2+4\text{x}+1$ increases on the interval:
  1. (1, 2)
  2. (2, 3)
  3. (1, 3)
  4. (2, 4)
✓

Answer

  1. (1, 3)
Solution:
Given, $\text{f}(\text{x})=2\log(\text{x}-2)-\text{x}^2+4\text{x}+1$
Domain of f(x) is $(2,\infty).$
$\text{f}'(\text{x})=\frac{2}{\text{x}-2}-2\text{x}+4$
$=\frac{2-2\text{x}^2+4\text{x}+4\text{x}-8}{\text{x}-2}$
$=\frac{-2\text{x}^2+8\text{x}-6}{\text{x}-2}$
$=\frac{-2(\text{x}^2-4\text{x}+3)}{\text{x}-2}$
For f(x) to be increasing, we must have
$\text{f}'(\text{x})>0$
$\Rightarrow\frac{-2(\text{x}^2-4\text{x}+3)}{\text{x}-2}>0$
$\Rightarrow\text{x}^2-4\text{x}+3<0$ $[\because\ \text{x}(\text{x}-2)>0\ \&-2<0]$
$\Rightarrow(\text{x}-1)(\text{x}-3)<0$
$\Rightarrow1<\text{x}<3$
$\Rightarrow\text{x}\in(1,3)$
Also, the domain of f(x) is $(2,\infty).$
$\Rightarrow\text{x}\in(1,3)\cap(2,\infty)$
$\Rightarrow\text{x}\in(1,3)$

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