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.

It indicates a horizontal tangent line, which could be a local maximum, local minimum, or saddle point.

The function $f(x)$ is increasing.

The function $f(x)$ is decreasing.

It indicates a possible point of inflection where the concavity may change.

The function $f(x)$ is concave up.

The function $f(x)$ is concave down.

Look for points where the function changes from increasing to decreasing.

Look for points where the function changes from decreasing to increasing.

It indicates that $f'(x)$ is undefined at that point.

The steeper the graph of $f(x)$, the larger the absolute value of $f'(x)$.

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).