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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions1 Mark
Question
Function $\text{f}(\text{x})=\log_\text{a}\text{x}$ is increasing on R, if:
  1. 0 < a < 1
  2. a > 1
  3. a < 1
  4. a > 0
✓

Answer

  1. a > 1

Solution:

$\text{f}(\text{x})=\log_\text{a}\text{x}=\frac{\log\text{x}}{\log\text{a}}$

$\text{f}'(\text{x})=\frac{1}{\text{x}\log\text{a}}$

Given: f(x) is increasing on R.

$\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in\text{R}$

$\Rightarrow\frac{1}{\text{x}\log\text{a}}>0,\forall\ \text{x}\in\text{R}$

$\Rightarrow\text{a}>1$

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