Application of Derivatives

$f^{\prime}(a), f^{\prime}(b), f^{\prime}(c)$ are in

$f^{\prime \prime}(a), f^{\prime \prime}(b), f^{\prime \prime}(c)$ are

If the sum of all the roots of the equation $e^{2x}-11e^{x}-45e^{-x}+\frac{81}{2}=0$is$\log\,_{e}P$, then $P$is equal to _____.

If$f\&g$are differentiable functions in$[0,1]$satisfying$f(0)=2=g(1),g(0)=0\&f(1)=6,$then for some$c∈]0, 1[$

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if }…

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number. Match each entry in List-I to the correct…

Let S be the set of all twice differentiable functions f from $ℝ$ to $ℝ$ such that $\frac{d^{2}f}{dx^{2}}(x)>0$ for all $x\in(-1,1)$. For $f\in S$, let $X_{f}$ be the number of points $x\in(-1,1)$ for which $f(x)=x$.…

Let $f:R\to R$ be defined by $f(x)=\frac{x^{2}-3x-6}{x^{2}+2x+4}$Then which of the following statements is(are) TRUE?

Consider all rectangles lying in the region ${(x,y)\in R\times R:0\leq x\leq\frac{\pi}{2}and0\leq y\leq2sin(2x)}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all…

Let $a,b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^{2}=4\lambda x$, and suppose the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through…

Let $f:R\to R$ be given by $f(x)=(x-1)(x-2)(x-5).$ Define $F(x)=\int0xf(t)dt,x>0.$ Then which of the following options is/are correct?

Let $f(x)=\frac{sin⁡\pi x}{x^{2}},x>0.$ Let $x_{1}<x_{2}<x_{3}<\ldots<x_{n}<\ldots$ be all the points of local maximum of $f$ and $y_{1}<y_{2}<y_{3}<\ldots<y_{n}<\ldots$ be all the points of local minimum of $f.$ Then…