If $f''(x) > 0$, the function is concave up. If $f''(x) < 0$, the function is concave down.

It uses the sign of the second derivative at critical points to determine if they are local maxima or minima.

At a local minimum, the function is concave up, resembling the bottom of a bowl, thus $f''(x) > 0$.

At a local maximum, the function is concave down, resembling the top of a hill, thus $f''(x) < 0$.

The test doesn't provide enough information to determine the nature of the critical point. The critical point may be a point of inflection, a local extremum, or neither.

When the second derivative is difficult to compute or when $f''(c) = 0$.

Extrema (local max or min) can only occur at critical points or endpoints of the interval.

Find where the first derivative is equal to zero or undefined.

The slope of the tangent line is increasing as x increases.

The slope of the tangent line is decreasing as x increases.

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

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 $f''(x)$.

Local: Max/min in a neighborhood. Global: Absolute max/min over the entire domain.

First: Uses the sign change of $f'(x)$ to determine extrema. Second: Uses the sign of $f''(x)$ at critical points to determine extrema.

Concavity: Refers to the rate of change of the slope. Increasing/Decreasing: Refers to whether the function's value is rising or falling.

Critical points: $f'(x) = 0$ or undefined. Inflection points: $f''(x)$ changes sign.

When the second derivative is easy to compute and non-zero at the critical point.

When the second derivative is difficult to compute or equals zero at the critical point.

Relative extrema: Local max or min within a specific interval. Absolute extrema: Global max or min over the entire domain.

Concave up: $f''(x) > 0$, curve opens upwards. Concave down: $f''(x) < 0$, curve opens downwards.

Critical points: Points where $f'(x) = 0$ or is undefined within the interval. Endpoints: Boundaries of the interval.

First derivative: Slope of the tangent line, rate of change. Second derivative: Concavity, rate of change of the slope.