Verified Solution • 2025–26 Session moderate

QJEEM23Macd49: JEE Main 2023 Mathematics - Class 12 JEEM23Macd49 Solution: If , then

If f(x)=x3−x2f′(1)+xf′′(2)−f′′′(3),x∈Rf(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R} , then

Multiple Choice Options

  • A 2f(0)−f(1)+f(3)=f(2)2f(0) - f(1) + f(3) = f(2)
  • B f(1)+f(2)+f(3)=f(0)f(1) + f(2) + f(3) = f(0)
  • C f(3)−f(2)=f(1)f(3) - f(2) = f(1)
  • D 3f(1)+f(2)=f(3)3f(1) + f(2) = f(3)

Step-by-Step Verified Solution

f(x)=x3−x2f′(1)+xf′′(2)−f′′′(3),x∈Rf(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R

Let f′(1)=a,f′′(2)=b,f′′′(3)=c\mathrm{f}^{\prime}(1)=\mathrm{a}, \mathrm{f}^{\prime \prime}(2)=\mathrm{b}, \mathrm{f}^{\prime \prime \prime}(3)=\mathrm{c}

amp;f(x)=x3−ax2+bx−camp;f′(x)=3x2−2ax+bamp;f′′(x)=6x−2aamp;f′′′(x)=6amp;c=6,a=3,b=6amp;f(x)=x3−3x2+6x−6amp;f(1)=−2,f(2)=2,f(3)=12,f(0)=−6amp;2f(0)−f(1)+f(3)=2=f(2)\begin{aligned}& f(x)=x^3-a x^2+b x-c \\\\& f^{\prime}(x)=3 x^2-2 a x+b \\\\& f^{\prime \prime}(x)=6 x-2 a \\\\& f^{\prime \prime \prime}(x)=6 \\\\& c=6, a=3, b=6 \\\\& f(x)=x^3-3 x^2+6 x-6 \\\\& f(1)=-2, f(2)=2, f(3)=12, f(0)=-6 \\\\& 2 f(0)-f(1)+f(3)=2=f(2)\end{aligned}

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