Differentiation & Applications · JEE Main previous year questions

2026 2 questions

1. Q112026 · 21 Jan 2026 · Shift 2

   Let $f:\mathbb{R}\to \mathbb{R}$ be a twice differentiable function such that $f^{\prime \prime }(x)>0$ for all $x\in \mathbb{R}$ and $f^{\prime }(a-1)=0$, where $a$ is a real number. Let $g(x)=f({\tan }^{2}x-2\tan x+a),0<x<\frac{\pi }{2}$. Consider the following two statements: (I) g is increasing in $\left(0,\frac{\pi }{4}\right)$ (II) g is decreasing in $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$ Then,

   1. A.Both (I) and (II) are True
   2. B.Neither (I) nor (II) is True
   3. C.Only (I) is True
   4. D.Only (II) is True
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   Answer: B

2. Q142026 · 21 Jan 2026 · Shift 2

   Let $f(x)=x^3+x^2{f}^{\prime }(1)+2x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3),x\in \mathbb{R}$. Then the value of $f^{\prime }(5)$ is :

   1. A.$\frac{657}{5}$
   2. B.$\frac{117}{5}$
   3. C.$\frac{2}{5}$
   4. D.$\frac{62}{5}$
   Show answer

   Answer: B