Critical points are key to understanding a function's behavior. They occur where the derivative is zero or undefined, indicating potential turning points or discontinuities in the graph.

The first derivative test helps determine if these points are local maxima, minima, or neither. By examining how the derivative's sign changes around critical points, we can identify peaks, valleys, and saddle points.

Critical Points and the First Derivative Test

Critical points of functions

• Points where the derivative is zero (stationary points) or undefined (non-differentiable points)
• Stationary points occur when the tangent line is horizontal ($f'(c) = 0$)
• Non-differentiable points arise from vertical tangents, cusps, or discontinuities ($f'(c)$ is undefined)
• Found by setting $f'(x) = 0$ and solving for $x$, then checking for any $x$-values that make $f'(x)$ undefined

First derivative test conditions

• Determines the nature of critical points and relative extrema for a continuous function $f$ on an open interval containing the critical point $c$
• If $f'$ changes from positive to negative at $c$, then $f(c)$ is a local maximum (peak)
• If $f'$ changes from negative to positive at $c$, then $f(c)$ is a local minimum (valley)
• If $f'$ does not change sign at $c$, then $f(c)$ is neither a local maximum nor a local minimum (saddle point or inflection point)

Nature of critical points

• Apply the first derivative test to determine if a critical point is a local maximum, local minimum, or neither

1. Find the critical points of the function by setting $f'(x) = 0$ and solving for $x$, and identifying any $x$-values that make $f'(x)$ undefined
2. Evaluate the sign of $f'(x)$ on the left and right sides of each critical point using test points
3. If the sign changes from positive to negative, the critical point is a local maximum
4. If the sign changes from negative to positive, the critical point is a local minimum
5. If the sign does not change, the critical point is neither a local maximum nor a local minimum

Relative extrema using derivatives

• Relative extrema are the local maxima and minima of a function
• Found by applying the first derivative test to each critical point
• If $f'$ changes from positive to negative at a critical point, it is a local maximum
• If $f'$ changes from negative to positive at a critical point, it is a local minimum
• The $y$-coordinates of the local maxima and minima are the relative maximum and minimum values of the function, respectively