Mean values theorem practice · Mixed difficulty

If Rolle's theorem holds for the function $f(x) 2 x^{3}+b x^{2}$ $+c x, x \in[-1,1]$, at the point $x=\frac{1}{2}$, then $2 b+c$ equals :

If $c$ is a point at which Rolle’s Theorem holds for the function, $f\left( x \right) = {\log _e}\left( {\frac{{{x^2} + \alpha }}{{7x}}} \right)$ in the interval $\left[ {3,\,4} \right]$ where $\alpha \, \in \,R,$ then $f''(c)$ is equal to

The value of c prescribed by Lagrange's mean value theorem, when $f(x)=\sqrt{x^2-4}, a=2$ and $b=3,$ is

If $a$ and $b$ are two distinct real roots of the polynomial $f\left( x \right)$ such that $a < b$, then there exists a real number $c$ lying between $a$ and $b$, such that

Let $f(x)=a x^5+b x^4+c x^3+d x^2+e x,$ where $a, b, c,$ $d, e \in R$ and $f(x)=0$ has a positive root $\alpha.$ Then,

If $f\left( x \right)$ is a function given by
$f\left( x \right) = \left| {\begin{array}{cccccccccccccccccccc} {\sin x}&{\sin a}&{\sin b}\\ {\cos x}&{\cos a}&{\cos b}\\ {\tan x}&{\tan a}&{\tan b} \end{array}} \right|,\mathrm{where\;}0 < a < b < \frac{\pi }{2}$
Then the equation $f'\left( x \right) = 0$

Let ${f}$ be a continuous function in $[0,3]$ and differentiable in $(0,3)$ such that ${f}(3)=0$. Then there exist some $\alpha \in(0,3)$ such that

The value of $c$ in the Lagrange’s mean value theorem for the function
$f(x) = \,{x^3} - 4{x^2} + 8x + 11,$ when $x \in \,[0,\,1]$ is

The value of c in (0, 2) satisfying the Mean value theorem for the function $f\left( x \right) = x{\left( {x - 1} \right)^2},\;x \in \left[ {0,\;2} \right]$ is equal to

Suppose $f$ is differentiable on $\mathbb{R}, f(0)=0$ and $1 \leq f^{\prime}(x) \leq 2$ for all $x \in \mathbb{R}$. Then

If [0, 1], Lagrange’s mean value theorem is NOT applicable to

Value of $c$ in the Lagrange's mean value theorem for the function

$f(x)=x+\frac{1}{x} \text { on }\left[\frac{1}{2}, 2\right\}$
is

If $f(x)=x^2-2 x+4$ and $\frac{f(5)-f(1)}{5-1}=f^{\prime}(c)$ then value of $\mathrm{c}$ will be

Consider a quadratic equation $a x^{2}+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=a \frac{x^{3}}{3}+b \frac{x^{2}}{2}+c x$.
Statement 1: The quadratic equation has at least one root in the interval $(0,1)$.
Statement 2: The Rolle's theorem is applicable to function $g(x)$ on the interval $[0,1]$.

Let a,b be two distinct roots of a polynomial $f\left( x \right)$. Then there exists at least one root lying between $a$ and $b$ of the polynomial

Let $S$ be the set of all functions $f:\left[ {0,\,1} \right] \to R,$ which are continuous on [0,1] and differentiable on (0,1). Then for every $f$ in $S$, there exists a $c\, \in (0,1),$ depending on $f$, such that

If Rolle’s theorem hold for the function $f\left( x \right) = {x^3} - a{x^2} + bx + 4,\,\,x \in \left[ {1,2} \right]$ with $f'\left( {\frac{4}{3}} \right) = 0$, then ordered pair $\left( {a,\,b} \right)$ is equal to

$f\left( x \right)$ satisfies the conditions of Rolle&rsquo;s theorem in [1, 2] and $f\left( x \right)$ is continuous in $\left[ {1,2} \right]$, then $\int \nolimits_1^2 f'\left( x \right)dx$ is equal to

If $a, b, c$ be non-zero real numbers such that $\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = 0,} }$ then the equation $a x^2+b x+c=0$ will have

If $f\left( x \right) = \left( {x - p} \right)\left( {x - q} \right)\left( {x - r} \right),$ where $p < q < r,$ are real numbers, then the application of Rolle's theorem of f leads to

If $2a + 3b + 6c = 0$, then at least one root of the equation $a{x^2} + bx + c = 0$ lies in the interval

If $f(x)=4^{\sin x}$ satisfies the Rolle's theorem on $[0, \pi]$, then the value of $c \in(0, \pi)$ for which $f^{\prime}(c)=0$ is

A function $f$ is defined by $f\left( x \right) = {e^x}\sin x$ in $\left[ {0,\;\pi } \right]$. Which of the following is not correct?

In the mean value theorem $f\left( b \right) - f\left( a \right) = \left( {b - a} \right)f'\left( c \right),\,if\,a = 4,\;b = 9$ and $f\left( x \right) = \sqrt x$, then the value of $c$ is

The position of a moving car at time $t$ is given by $f(t)=a t^2+b t+c, t>0$ where $a, b$ and $c$ are real numbers greater than 1. Then the average speed of the car over the interval $\left[t_1, t_2\right]$ is attained at the point

If $f(1)=3$ and $f^{\prime}({x}) \leq 1.4$ for $1 \leq {x} \leq 8$. The largest possible value which ${f} (8)$ can have, is