Application of Derivatives — JEE Advanced PYQs

If a continuous function $f$ defined on the real line $\mathbf{R}$, assumes positive and negative values in $\mathbf{R}$ then the equation $f(x)=0$ has a root in $\mathbf{R}$. For example, if it is known that a continuous function $f$ on $\mathbf{R}$ is positive at …

Let $\theta \in \left(0,\frac{\pi }{4}\right)$ and $t_1=(\tan \theta {)}^{\tan \theta },t_2=(\tan \theta {)}^{\cot \theta },t_3=(\cot \theta {)}^{\tan \theta }$ and $t_4=(\cot \theta {)}^{\cot \theta }$, then

f(x) is cubic polynomial which has local maximum at $x=-1$. If $f(2)=18,f(1)=-1$ and $f^{\prime }(x)$ has local minima at $x=0$, then

$f(x)=\{\begin{array}{ll}{e}^x,& 0\le x\le 1\\ 2-e^{x-1},& 1<x\le 2\\ x-e,& 2<x\le 3\end{array}$ and $g(x)={\int }_0^{x}f(t)dt,x\in [1,3]$ then $g(x)$ has

If $\mathrm{f}(\mathrm{x})$ is a twice differentiable function such that $\mathrm{f}(\mathrm{a})=0,\mathrm{f}(\mathrm{b})=2,\mathrm{f}(\mathrm{c})=-1,\mathrm{f}(\mathrm{d})=2,\mathrm{f}(\mathrm{e})=0$, where $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}<\mathrm{e}$, then …