JEE Main · 2005 · 08 May 2005 · Offline · Application of Derivatives

More JEE Main questions on Application of Derivatives

1. Let $f:R\to R$ be a twice differentiable function such that the quadratic equation $\mathrm{f}(\mathrm{x}){\mathrm{m}}^2-2{\mathrm{f}}^{\prime }(\mathrm{x})\mathrm{m}+{\mathrm{f}}^{\prime }(\mathrm{x})=0$ in m , has two equal roots for every …
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2. Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that $f^{\prime \prime }(x)>0$ for all $x\in \mathbf{R}$ and $f^{\prime }(a-1)=0$, where a is real number. Let $g(x)=f\left({\tan }^{2}x-2\tan x+a\right)$, $0<x<\frac{\pi }{2}$.
   
   Consider the following two statements:
   (I) g i …
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3. Let the maximum value of ${\left({\sin }^{-1}\mathrm{x}\right)}^2+{\left({\cos }^{-1}\mathrm{x}\right)}^2$ for $x\in \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n}{\pi }^2$, where $\gcd (m,n)=1$.
   
   Then $\mathrm{m}+\mathrm{n}$ is equal to ____.
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4. Let $f(x)=x^{2025}-x^{2000},x\in [0,1]$ and the minimum value of the function $\mathrm{f}(\mathrm{x})$ in the interval $[0,1]$ be $(80{)}^{80}(\mathrm{n}{)}^{-81}$. Then n is equal to
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5. Let $(2\alpha ,\alpha )$ be the largest interval in which the function $f(t)=\frac{|t+1|}{t^2},t<0$, is strictly decreasing. Then the local maximum value of the function
   $g(x)=2{\log }_e(x-2)+\alpha x^2+4x-\alpha ,x>2$, is ______
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6. Let f be a differentiable function satisfying
   $\mathrm{f}(\mathrm{x})=1-2\mathrm{x}+{\int }_0^{\mathrm{x}}{\mathrm{e}}^{(\mathrm{x}-\mathrm{t})}\mathrm{f}(\mathrm{t})\mathrm{dt},\mathrm{x}\in \mathbf{R}$ and let …
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