Learning Outcomes

Upon completing this lesson, you will be able to:

• Define critical points and understand their significance in finding extrema.
• Use the First Derivative Test to identify intervals where a function is increasing or decreasing and to classify local maxima and minima.
• Define concavity and use the Second Derivative Test to identify intervals where a function is concave up or concave down.
• Define and locate points of inflection where the concavity of a function changes.
• Combine these analytical tools to create an accurate sketch of a function's graph.

Topics Covered

• Finding Maxima and Minima
• Concavity & Points of Inflection

I. Finding Maxima and Minima

One of the most powerful applications of the derivative is in finding the maximum and minimum values of a function, known as its extrema. These points often correspond to the most important features of a graph, such as the peak of a trajectory or the lowest cost of production.

A. Critical Points

Local maxima and minima can only occur at points where the slope of the tangent line is either horizontal or does not exist. These special points are called critical points.

A critical point of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ is undefined.

To find the critical points of a function, you must:

1. Find the first derivative, $f'(x)$.
2. Set the derivative equal to zero ($f'(x) = 0$) and solve for $x$. These are the points with horizontal tangents.
3. Find any $x$-values for which the derivative is undefined (e.g., where the denominator of $f'(x)$ is zero).

B. The First Derivative Test

Once we have the critical points, the First Derivative Test allows us to determine whether each point is a local maximum, a local minimum, or neither. The test relies on the following principle:

• If $f'(x) > 0$ on an interval, then f is increasing on that interval.
• If $f'(x) < 0$ on an interval, then f is decreasing on that interval.

The test is performed as follows:

1. Find all critical points of the function and use them to divide the number line into intervals.
2. Choose a test value from within each interval and plug it into the first derivative, $f'(x)$, to determine its sign ($+$ or $-$).
3. Analyze the sign changes at each critical point $c$:
   • If $f'(x)$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$.
   • If $f'(x)$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.
   • If $f'(x)$ does not change sign at $c$, then there is neither a maximum nor a minimum at $c$.

Example 1