The first derivative of a function reveals its behavior, showing where it increases, decreases, or stays constant. By analyzing the sign of f'(x), we can determine how f(x) changes across different intervals.

Critical points are key values where a function's behavior shifts. By finding where f'(x) equals zero or is undefined, we can identify these points and classify them as local maxima, minima, or neither using derivative tests.

Increasing and Decreasing Functions

First derivative for function behavior

• Determines whether a function $f(x)$ is increasing or decreasing on an interval based on the sign of its first derivative $f'(x)$
• Function $f(x)$ is increasing on an interval when $f'(x) > 0$ for all $x$ in that interval, meaning as $x$ increases, $f(x)$ also increases (positive slope)
• Function $f(x)$ is decreasing on an interval when $f'(x) < 0$ for all $x$ in that interval, meaning as $x$ increases, $f(x)$ decreases (negative slope)
• Function $f(x)$ is constant on an interval when $f'(x) = 0$ for all $x$ in that interval, meaning the function maintains the same value (zero slope)

Intervals of function change

• Find the first derivative of the function, denoted as $f'(x)$, to analyze its behavior
• Solve $f'(x) = 0$ to find critical points where the function changes from increasing to decreasing or vice versa
• Evaluate the sign of $f'(x)$ on the intervals between critical points to determine whether the function is increasing or decreasing on each interval
  • $f'(x) > 0$ on an interval indicates the function is increasing on that interval
  • $f'(x) < 0$ on an interval indicates the function is decreasing on that interval
• Utilize a sign chart or number line to visualize and clearly represent the intervals of increase and decrease

Real-world applications of function behavior

• Identify the function that accurately models the real-world situation (position, velocity, population growth)
• Analyze the intervals where the function is increasing or decreasing to understand the behavior of the modeled situation
• Interpret the meaning of increasing and decreasing intervals in the context of the problem
  • Height of a projectile over time: increasing intervals represent the projectile rising, while decreasing intervals represent the projectile falling
  • Population growth: increasing intervals indicate population expansion, while decreasing intervals indicate population decline

Critical Points of Functions

Critical points of functions

• Critical points are values of $x$ where the first derivative $f'(x)$ equals zero or does not exist (undefined)
• Find critical points by solving $f'(x) = 0$ for $x$ and identifying values of $x$ where $f'(x)$ is undefined
• Classify critical points as local maximum, local minimum, or neither using the following tests:
  1. First derivative test:
     • Local maximum: $f'(x)$ changes from positive to negative at the critical point
     • Local minimum: $f'(x)$ changes from negative to positive at the critical point
     • Neither: $f'(x)$ does not change sign at the critical point
  2. Second derivative test (applicable if $f''(x)$ exists):
     • Local maximum: $f'(x) = 0$ and $f''(x) < 0$ at the critical point
     • Local minimum: $f'(x) = 0$ and $f''(x) > 0$ at the critical point
     • Inconclusive: $f'(x) = 0$ and $f''(x) = 0$ at the critical point