From mean value theorem $f(b) - f(a) = $ $(b - a)f'({x_1});$ $a < {x_1} < b$ if $f(x) = {1 \over x}$, then ${x_1} = $
$\sqrt {ab} $
${{a + b} \over 2}$
${{2ab} \over {a + b}}$
${{b - a} \over {b + a}}$
If $f$ and $g$ are differentiable functions in $[0, 1]$ satisfying $f\left( 0 \right) = 2 = g\left( 1 \right)\;,\;\;g\left( 0 \right) = 0,$ and $f\left( 1 \right) = 6,$ then for some $c \in \left] {0,1} \right[$ . .
The abscissa of the points of the curve $y = {x^3}$ in the interval $ [-2, 2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[-2, 2], $ are
If the function $f(x) = 2x^2 + 3x + 5$ satisfies $LMVT$ at $x = 3$ on the closed interval $[1, a]$ then the value of $a$ is equal to
If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a twice differentiable function such that $f^{\prime \prime}(x)>0$ for all $x \in \mathrm{R}$, and $f\left(\frac{1}{2}\right)=\frac{1}{2}, f(1)=1$, then
Rolle's theorem is not applicable to the function $f(x) = |x|$ defined on $ [-1, 1] $ because