Mean values theorem practice · Easy

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

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

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

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

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