1. Find critical points. 2. Compute the second derivative. 3. Evaluate the second derivative at each critical point. 4. Determine if each point is a local max, min, or neither.

1. Find the second derivative, $f''(x)$. 2. Set $f''(x) = 0$ and solve for x. 3. Create a sign chart for $f''(x)$. 4. Determine intervals of concave up ($f''(x) > 0$) and concave down ($f''(x) < 0$).

1. Find the second derivative, $f''(x)$. 2. Set $f''(x) = 0$ and solve for x. 3. Check if the concavity changes at each potential inflection point.

Use the First Derivative Test or analyze the behavior of the function around the critical point.

Find $f'(x)$, set $f'(c) = 0$ to confirm c is a critical point. Then, find $f''(x)$ and evaluate $f''(c)$. If $f''(c) < 0$, then $x = c$ is a local max.

Find $f'(x)$, set $f'(c) = 0$ to confirm c is a critical point. Then, find $f''(x)$ and evaluate $f''(c)$. If $f''(c) > 0$, then $x = c$ is a local min.

If the function is continuous and has only one critical point, and that point is a local extremum, then it's also the global extremum.

1. Find $f'(x) = 4cos(x)$. 2. Set $4cos(x) = 0$. 3. Solve for x, which gives $x = \frac{\pi}{2}, \frac{3\pi}{2}$.

1. Find $f''(x) = -4sin(x)$. 2. Evaluate $f''(\frac{\pi}{2}) = -4$. 3. Since $f''(\frac{\pi}{2}) < 0$, it's a local max.

1. Set $f'(x) = 0$ to find critical points, but there are none. 2. Check where $f'(x)$ is undefined, which is at $x=0$. 3. Since $f(x)$ is not defined at $x=0$, there are no local extrema.

$f''(x) = \frac{d^2}{dx^2}f(x)$

Solve $f'(x) = 0$ or find where $f'(x)$ is undefined.

If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local minimum.

If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local maximum.

$f'(x) = 2x^2 - 5x - 3$, $f''(x) = 4x - 5$

$f'(x) = 4cos(x)$, $f''(x) = -4sin(x)$

$f'(x) = \frac{494}{x}$, $f''(x) = -\frac{494}{x^2}$

Check the sign of $f''(x)$.

Solve $f''(x) = 0$ or find where $f''(x)$ is undefined.

1. Find critical points using $f'(x)$. 2. Plug critical points into $f''(x)$. 3. Determine local min/max based on the sign of $f''(x)$.

A point where f'(x) = 0 or f'(x) does not exist.

A point where the function's value is greater than or equal to the values at all nearby points.

A point where the function's value is less than or equal to the values at all nearby points.

The direction in which a curve bends. Concave up means the curve opens upwards; concave down means it opens downwards.

A point on a curve where the concavity changes (from up to down or vice versa).

The concavity of the function and helps determine local extrema.

A method using the second derivative to determine whether a critical point is a local maximum or minimum.

The function is concave up.

The function is concave down.

The test fails to determine whether the critical point is a local max, min, or neither. Further analysis is needed.