Verified Solution • 2025–26 Session moderate

QJEEM23Mad69d: JEE Main 2023 Mathematics - Class 12 JEEM23Mad69d Solution: If , then is equal to

If y(x)=x^{x},x > 0 , then y′′(2)−2y′(2)y''(2)-2y'(2) is equal to

Multiple Choice Options

  • A 4(log⁡e2)2+24(\log_{e}2)^{2}+2
  • B 8log⁡e2−28\log_{e}2-2
  • C 4log⁡e2+24\log_{e}2+2
  • D 4(log⁡e2)2−24(\log_{e}2)^{2}-2

Step-by-Step Verified Solution

amp;y=xxamp;y′=xx(1+ln⁡x)amp;y′′=xx(1+ln⁡x)2+xxxamp;f′′(2)−2f′(2)=(4(1+ln⁡2)2+2)−(2)(4(1+ln⁡2))amp;=4(1+(ln⁡2)2)+2−8amp;=4(ln⁡2)2−2amp;\begin{aligned} & y=x^x \\\\ & y^{\prime}=x^x(1+\ln x) \\\\ & y^{\prime \prime}=x^x(1+\ln x)^2+\frac{x^x}{x} \\\\ & f^{\prime \prime}(2)-2 f^{\prime}(2)=\left(4(1+\ln 2)^2+2\right)-(2)(4(1+\ln 2)) \\\\ & =4\left(1+(\ln 2)^2\right)+2-8 \\\\ & =4(\ln 2)^2-2 \\\\ & \end{aligned}

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