Analytical Applications of Differentiation

There exists an inflection point at $(c,f(c))$.

The second derivative test yields $f''(c) > 0$ at this critical point.

Function $f(x)$ is decreasing on both sides of $c$ in its domain.

The first derivative test shows that $f'(c)$ changes from negative to positive.

They occur only at endpoints of function g(t)'s domain.

They are always global extrema for function $g(t)$ on its domain.

They are points where the second derivative test fails.

They must yield either relative maxima or minima on the interval.

Maximizing area

Maximizing volume

Minimizing cost

Minimizing area

The problem is unsolvable.

There is no restriction on the function's values.

The problem has multiple solutions.

The function is not continuous.

$h(x)= x^7 - \frac{x^9}{9}$

$h(x)= -7x + \frac{1}{6}e^{3x}$

$h(x)= -7x - \frac{x^3}{6}$

$h(x)= e^{-7x} + x^3$

The point where the function has its lowest value.

The point where the function is undefined.

The point where the function has its highest or lowest value.

The point where the function intersects the x-axis.

Points where the function reaches the maximum or minimum values within a specific interval.

The overall maximum and minimum values of the function.

Points where the function has the highest or lowest values within the entire domain.

Points where the function is undefined.

$f'(a) > 0$

The value of $f'(a)$ cannot be determined.

$f'(a) = 0$

$f'(a) < 0$

The function's slope is non-decreasing (i.e., its first derivative does not decrease).

The function exhibits symmetry across its axis of minimum value in $(a,b)$.

There exists some subinterval in which the second derivative of the function is always negative.

The function's slope is constant throughout $(a,b)$.

Identifying the critical points.

Finding the absolute extrema.

Evaluating the function at the endpoints of the interval.

Reading the problem carefully and identifying the equation and constraint.