Analytical Applications of Differentiation

The function $f(x)$ must be continuous on the open interval $(a, b)$.

The function $f(x)$ must be differentiable on the closed interval $[a, b]$.

The function $f(x)$ must be continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$.

The function $f(x)$ must be differentiable everywhere.

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The Mean Value Theorem

The Extreme Value Theorem

Rolle's Theorem

The Intermediate Value Theorem

$f(x)$ is differentiable on $[a, b]$.

$f(x)$ is continuous on $[a, b]$.

$f(x)$ is continuous on $(a, b)$.

$f(x)$ is differentiable on $(a, b)$.

A point where the function reaches its absolute highest value on the entire domain.

A point where the function reaches its absolute lowest value on the entire domain.

A point where the function changes direction, forming a peak or a valley within a specific subinterval.

An endpoint of the interval $[a, b]$.

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$f(x)$ is decreasing.

$f(x)$ is increasing.

$f(x)$ is concave up.

$f(x)$ is concave down.

(-1, 1)

$(-\infty, -1)$ and $(1, \infty)$

(-$\infty$, 0) and (0, $\infty$)

(-$\infty$, $\infty$)

f(x) is differentiable at x=c

f(x) has a local extremum at x=c

f(x) is neither increasing nor decreasing at x=c

f(x) is continuous at x=c

$f(x)$ is concave up.

$f(x)$ is concave down.

$f(x)$ is increasing.

$f(x)$ is decreasing.