1. Find $f'(x)$. 2. Find critical points. 3. Test intervals between critical points in $f'(x)$. 4. Determine if $f'(x)$ is positive (increasing) or negative (decreasing).

1. Find critical points. 2. Determine intervals of increasing/decreasing. 3. If $f'(x)$ changes from + to -, local max. If $f'(x)$ changes from - to +, local min.

1. Find $f''(x)$. 2. Find possible points of inflection where $f''(x) = 0$ or is undefined. 3. Check if concavity changes at these points.

1. Find $f''(x)$. 2. Find where $f''(x) = 0$ or is undefined. 3. Test intervals to determine if $f''(x)$ is positive (concave up) or negative (concave down).

1. Find the domain and discontinuities. 2. Find intercepts and symmetry. 3. Find critical points. 4. Determine increasing/decreasing intervals. 5. Find extrema. 6. Determine concavity and points of inflection. 7. Sketch the graph.

1. Find critical points. 2. Find $f''(x)$. 3. Evaluate $f''(x)$ at each critical point. 4. If $f''(c) > 0$, local min. If $f''(c) < 0$, local max.

1. Find $f(-x)$. 2. If $f(-x) = f(x)$, the function is even. 3. If $f(-x) = -f(x)$, the function is odd.

1. Identify values of $x$ that make the denominator zero. 2. Exclude those values from the set of all real numbers.

The domain is all real numbers unless there are specific restrictions given.

1. Set $f(x) = 0$. 2. Solve for $x$. 3. The solutions are the x-intercepts.

If $f'(x) > 0$, $f(x)$ is increasing. If $f'(x) < 0$, $f(x)$ is decreasing.

If $f'(x)$ changes from positive to negative at $x=c$, then $f(x)$ has a local maximum at $x=c$. If $f'(x)$ changes from negative to positive at $x=c$, then $f(x)$ has a local minimum at $x=c$.

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

Find where $f''(x) = 0$ or is undefined, and check if the concavity changes at those points.

Check if $f(-x) = f(x)$. If true, the function is even and symmetric about the y-axis.

Check if $f(-x) = -f(x)$. If true, the function is odd and symmetric about the origin.

A point where the function is not continuous, possibly indicating a hole, jump, or vertical asymptote.

Critical points are potential locations of local maxima, local minima, or saddle points of a function.

Uses the second derivative to determine if a critical point is a local max or min. If $f''(c) > 0$, local minimum. If $f''(c) < 0$, local maximum.

All real numbers, unless otherwise restricted.

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

First Derivative Test: uses sign changes of $f'(x)$ to find extrema. Second Derivative Test: uses the sign of $f''(x)$ at critical points to find extrema.

Even functions: symmetric about the y-axis, $f(-x) = f(x)$. Odd functions: symmetric about the origin, $f(-x) = -f(x)$.

Increasing: $f'(x) > 0$. Concave Up: $f''(x) > 0$. A function can be increasing and concave down, or increasing and concave up.

Decreasing: $f'(x) < 0$. Concave Down: $f''(x) < 0$. A function can be decreasing and concave up, or decreasing and concave down.

Critical Point: $f'(x) = 0$ or undefined (potential max/min). Point of Inflection: $f''(x) = 0$ or undefined (concavity change).

Root: where the function crosses the x-axis. Y-intercept: where the function crosses the y-axis.

Critical points of $f(x)$: $f'(x) = 0$ or undefined. Critical points of $f'(x)$: $f''(x) = 0$ or undefined.

Even symmetry: symmetric about the y-axis. Odd symmetry: symmetric about the origin.

Zero: Find where the function equals zero. Undefined: Find where the function has a discontinuity (e.g., division by zero).