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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Continuity and Differentiability

NTA Abhyas 2020 · Mathematics · Continuity and Differentiability

If f (x) = [x] ^ \ x\ + \ x\ ^ [x] + ( ), where [·] and · represent...

If ff\left(x\right) = {\left[x\right]}^{\left\{x\right\}} + {\left\{x\right\}}^{\left[x\right]} + \sin \left(\pix\right),where[⋅]and⋅representthegreatestintegerfunctionandthefractionalpartfunctionrespectively,thenwhere [·] and {·} represent the greatest integer function and the fractional part function respectively, thenf^{'} \left(\dfrac{7}{2}\right)is equal to

Options

  1. A3ln⁡3+π4\sqrt{3}\ln 3 + \dfrac{\pi}{4}
  2. B3ln⁡3+3π4\sqrt{3}\ln 3 + \dfrac{3\pi}{4}
  3. C3ln⁡3+π+34\sqrt{3}\ln 3 + \pi + \dfrac{3}{4}
  4. D3ln⁡3+34\sqrt{3}\ln 3 + \dfrac{3}{4}

Correct answer

D. 3ln⁡3+34\sqrt{3}\ln 3 + \dfrac{3}{4}

Step-by-step solution

& f(x)=[x]{x}+{x}f\left(x\right) = { \left[x\right]}^{ \left\{x\right\}} + {\left\{x\right\}}}^{\left[x\right]} + \sin \left(\pix\right)<br />f \left(x\right) = {3}^{x - 3} + { \left(x - 3\right)}^{3} + \sin \left(\pix\right),x∈(3,4)<br/>x \in \left(3, 4\right)<br />f^{'} \left(x\right)=3x−3ln⁡3+3(x−3)2+πcos= {3}^{x - 3}\ln 3 + 3 \left(x - 3\right)2 + \pi cos\left(\pix\right)
∴ f^{'}\left(\dfrac{7}{2}\right)=3ln⁡3+34= \sqrt{3}\ln 3 + \dfrac{3}{4}
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