Consider the function $f(x) = e^{-2x} \sin 2x$ over the interval $(0, \pi/2)$. $A$ real number $c \in (0, \pi/2)$,as guaranteed by Rolle's theorem,such that $f'(c) = 0$ is

Consider the function $f(x) = |x - 2| + |x - 5|$,$x \in R$.
Statement-$1$: $f'(4) = 0$.
Statement-$2$: $f$ is continuous in $[2, 5]$,differentiable in $(2, 5)$,and $f(2) = f(5)$.

For the curve $y = x^3$ in the interval $[-2, 2]$,find the abscissae of the points where the slope of the tangent is equal to the slope of the secant line passing through the endpoints of the interval,as per the Mean Value Theorem.

If from the Mean Value Theorem,$f'({x_1}) = \frac{f(b) - f(a)}{b - a}$,then

$f:[2,10] \rightarrow R$ is defined as $f(x) = \begin{cases} \frac{1}{2}(x-6)^2-3, & x \leq 4 \\ x-5, & x > 4 \end{cases}$. Which of the following is true?

Let $y = f(x)$ and $y = g(x)$ be two differentiable functions in $[0, 2]$ such that $f(0) = 3$,$f(2) = 5$,$g(0) = 1$,and $g(2) = 2$. If there exists at least one $c \in (0, 2)$ such that $f'(c) = k g'(c)$,then $k$ must be: