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R,
Question: Functionf(x)=\log _{a}xis increasing on
(a)0<a<1
a>1
a<1
a>0
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R,
Question: Functionf(x)=\log _{a}xis increasing on
(a)0<a<1
a>1
a<1
a>0

Answer

Solution: To solve this question, first, we need to find the first derivative of the given function and apply the condition for an increasing functionf'(x)>0,We have,f(x)=\log _{a}xf(x)=\frac {\log x}{\lg a} Differentiate the above function with respect to x.f'(x)=\frac {d}{dx}(\frac {\log g}{la}a)f'(x)=\frac {1}{\log a}\frac {d}{\Delta x}(lxgx)f'(x)=\frac {1}{\log a}\times \frac {1}{x}f'(x)=\frac {1}{x\ln a} For a function to be increasing,f'(x)>0,\forall x\in Rxloga > 0,Vx∈R
For\frac {1}{x\lg a}always greater than0,x\log a>0=a>1Hence, option (b) is the correct answer.
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