JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Application of Derivatives

More JEE Advanced questions on Application of Derivatives

1. Consider the function $f:(0,\infty )\to (-\infty ,\infty )$ given by
   

   $$f(\mathrm{x})=\sqrt{\mathrm{x}}{\log }_{\mathrm{e}}(\mathrm{x})-\mathrm{x}+1$$

   
   Then which one of the following statements is TRUE ?
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2. Let P be the point on the parabola $\mathrm{y}={\mathrm{x}}^2$ such that the slope of the tangent to the parabola at the point P is 4. Let Q be the point in the first quadrant lying on the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=2$ such that the slope of the tangent to the circ …
   JEE Advanced 2026 Copy linkView solution →
3. Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f:\mathbb{R}\to \mathbb{R}$ by
   

   $$f(\mathrm{x})=\{\begin{array}{cc}2-2x^2-x^2\sin \frac{1}{x}& \text{ if }x\ne 0\\ 2& \text{ if }x=0\end{array}$$

   
   Then which one o …
   JEE Advanced 2025 Copy linkView solution →
4. Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be defined by
   

   $$f(x)=\{\begin{array}{ll}\frac{6x+\sin x}{2x+\sin x}& \text{ if }x\ne 0,\\ \frac{7}{3}& \text{ if }x=0.\end{array}$$

   
   Then which of the following statements …
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5. Let the function $f:\mathbb{R}\to \mathbb{R}$ be defined by
   

   $$f(x)=\frac{\sin x}{e^{\pi x}}\frac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}}\frac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}.$$

   
   Then the number of solut …
   JEE Advanced 2024 Copy linkView solution →
6. Let S be the set of all twice differentiable functions f from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{{\mathrm{d}}^2\mathrm{f}}{{\mathrm{dx}}^2}(\mathrm{x})>0$ for all $\mathrm{x}\in (-1,1)$. For $\mathrm{f}\in \mathrm{S}$, let ${\mathrm{X}}_{\mathrm{f}}$ be the number of p …
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