Analytical Applications of Differentiation

There exists at least one number $c$ in $(a, b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

There exists at least one number $c$ in $(a, b)$ such that $\frac{f(b)-f(a)}{b-a} = 0$.

There exists at least one number $c$ in $(a, b)$ such that $f''(c) = 0$.

There exists at least one number $c$ in $(a, b)$ such that $f(c) = f(a)$.

Function h is continuous on [1,e] and differentiable on (1,e).

Function h has equal values at endpoints x=1 and x=e.

Function h achieves its maximum value somewhere within interval [1,e].

Function h has no critical points within interval (1,e).

Each having zero as their minimum value somewhere between a and b.

Both having reciprocal first derivatives throughout $[a,b]$.

One of them displaying constant rate of change across any subinterval within $[a,b]$.

Both functions being continuous on $[a,b]$ and differentiable on $(a,b)$.

Their first derivatives must intersect exactly once within $(a,b)$.

Both functions must have identical concavity on $(a,b)$.

Their derivatives must be equal for all $x$ in $(a,b)$.

Their derivatives need not have matching values for all $x$ in $(a,b)$, only guaranteed at least once.

There exists a value $c$ in $(0, 2)$ such that $g'(c) = \frac{e^2 - 1}{2}$.

There exists a value $c$ in $[0, 2]$ such that $g'(c) = 0$.

There exists a value $c$ in $[0, 2]$ such that $g(c) = \frac{e^2 - 1}{2}$.

There exists a value $c$ in $(0, 2)$ such that $g'(c) = e$.

$\frac{f(b) + f(a)}{2}$

$\frac{f(b) - f(a)}{b - a}$

$f(b) - f(a)$

$\frac{b - a}{f(b) - f(a)}$

Infinitely many values of c exist.

Two distinct values of c exist.

Exactly one value of c exists.

No values of c satisfy this equation.

No C exists in (-3,7) as P''(x) has a constant slope of 0

Every x in (-3,7), the P''(x) = $\frac{P(7) - P(-3)}{14}$

There are two distinct numbers C in (-3,7) such that P''(C) = $\frac{P(7) - P(-3)}{14}$

There is at least one C in (-3,7) such that P'(C) = $\frac{P(7) - P(-3)}{14}$

There exists a value $c$ in $(0, \\pi]$ such that $h(c) = 1$.

There exists a value $c$ in $(0, \\pi)$ such that $h'(c) = 1$.

There exists a value $c$ in $[0, \\pi]$ such that $h'(c) = -1$.

There exists a value $c$ in $(0, \\pi)$ such that $h'(c) = 0$.

The function's second derivative, $f''(x)$, is zero or does not exist at $x = c$.

The function has no relative extrema on the interval despite having an inflection point.

The function's second derivative, $f''(x)$, changes sign at some point other than $x = c$.

There are multiple points where $f'(x) = \frac{f(b) - f(a)}{b - a}$ even if there's only one inflection point.