Analytical Applications of Differentiation

Yes, a global extremum is also a local extremum.

Yes, but only if the local extremum occurs at x = 1.

Yes, but only if the local extremum occurs at x = 0.

No, a local extremum can never be a global extremum.

$x = -1, x = 1$

$x = 0$

$x = 2$

$x = -2$

The absence of the derivative at $x = c$ guarantees no extreme value can occur there.

A critical point may occur at $x = c$, potentially leading to an absolute extremum if all other values are less extreme.

The Extreme Value Theorem ensures that an absolute maximum and minimum cannot exist in this scenario.

Since the derivative does not exist at $x = c$, it implies that both an absolute maximum and minimum must occur at points different from $c$.

$x = 0$

$x = 3$

$x = 1$

$x = 2$

The value of $f(c)$ is greater than or equal to every other function value on the interval.

The second derivative of $f(x)$ is positive at $x = c$.

The derivative of $f(x)$ is zero at $x = c$.

There are no other critical points in the interval.

The first derivative test shows that $f'(x)$ decreases before and after $p$.

The original function, $f(p)$, is greater than zero.

The second derivative, $f''(p)$, equals zero and changes sign around $p$.

The first derivative, $f'(p)$, equals zero but does not change sign around $p$.

Function $f$ reaches global maxima always when $x=c$.

There's likely to be local minima at $x=c$.

There should exist an inflection point exactly when $x=c$.

Existence of critical point at $x=c$ implies discontinuity in $f$.

Endpoints guarantee only local extrema while critical points provide information about global extrema as well.

Evaluating functions at endpoints may give false positives for global extrema due to potential discontinuities there.

Endpoints do not exist for open intervals; thus we rely on critical points to identify possible global extrema within these intervals.

Critical points are easier to calculate than evaluating functions at their endpoints for any given interval.

Points where the function is always increasing

Points where the function has an asymptote

Points where the function is always decreasing

Points where the function is not differentiable or where the derivative is equal to 0

By analyzing the rate of change of the function

By analyzing the function values at the endpoints of the interval

By analyzing the integral of the function

By analyzing the signs and changes in the first and second derivatives