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BITSAT 2012
Exam
BITSAT
Year
2012
Subject
Mathematics
Chapter
Application of Derivatives

BITSAT 2012 · Mathematics · Application of Derivatives

If f(x)=x^ α x and f(0)=0, then the value of α for which Rolle's...

If f(x)=xαlog⁡xf(x)=x^{\alpha} \log x and f(0)=0,f(0)=0, then the value of α\alpha for which Rolle's theorem can be applied in [0,1] is

Options

  1. A-2
  2. B-1
  3. C0
  4. D1/21 / 2

Correct answer

D. 1/21 / 2

Step-by-step solution

For Rolle's theorem in [a,b],f(a)=f(b),In⁡[0,1]⇒f(0)=f(1)=0∵[\mathrm{a}, \mathrm{b}], \mathrm{f}(\mathrm{a})=\mathrm{f}(\mathrm{b}),\operatorname{In}[0,1] \Rightarrow \mathrm{f}(0)=\mathrm{f}(1)=0\because the function has to be continuous in [0,1] ⇒f(0)=lim⁡x→0+f(x)=0⇒lim⁡x→0xαlog⁡x=0⇒lim⁡x→0log⁡xx−α=0\Rightarrow f(0)=\lim _{x \rightarrow 0^{+}} f(x)=0\Rightarrow \lim _{x \rightarrow 0} x^{\alpha} \log x=0 \Rightarrow \lim _{x \rightarrow 0} \frac{\log x}{x^{-\alpha}}=0 Applying L.H. Rule lim⁡x→01/x−αx−α−1=0⇒lim⁡x→0−xαα=0⇒α>0\lim _{x \rightarrow 0} \frac{1 / x}{-\alpha x^{-\alpha-1}}=0\Rightarrow \lim _{x \rightarrow 0} \frac{-x^{\alpha}}{\alpha}=0 \Rightarrow \alpha > 0
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