Lecture #18: Extrema and Graphs of Functions
MATH 145 – Calculus I
October 27, 2025
\(f'(x)\), \(f''(x)\) and the graph of \(f(x)\)
Suppose we want to sketch the graph of \(f(x) = x^2 + x - 12\).
We could plug in numbers for \(x\) and then plot those points until we can draw the graph.
However, we can use what we know about derivatives to make it easier.
\(f'(x)\), \(f''(x)\) and the graph of \(f(x)\)
Recall that:
- If \(f'(x) > 0\), then \(f(x)\) is increasing.
- If \(f'(x) < 0\), then \(f(x)\) is decreasing.
- If \(f''(x) > 0\), then \(f(x)\) is concave up.
- If \(f''(x) < 0\), then \(f(x)\) is concave down.
What happens when \(f'(x) = 0\) or \(f''(x) = 0\)?
Critical Points
If \(f(x)\) is a function, a critical point of \(f(x)\) is a value \(p\) such that \(f'(p) = 0,\) or undefined. The critical value is the value of \(f(p)\).
A local minimum or a local maximum of \(f(x)\) may exist at a critical point of \(f(x)\), so we can identify potential local extrema of \(f(x)\) by solving the equation \(f'(x) = 0\).
The First Derivative Test for Local Extrema
Given a critical point \(p\) of a function \(f(x)\), as you move left to right:
- if \(f'(x)\) switches from negative to positive, then \(f(x)\) has a local minimum there.
- if \(f'(x)\) switches from positive to negative, then \(f(x)\) has a local maximum there.
Example: The First Derivative Test
Example: Find all critical points of the function \(f(x) = x^3 - 6x^2 + 9x\). Sketch a graph.
Example: Find all critical points of the function \(f(x) = x^4 - 8x^2\). Sketch a graph.
The Second Derivative Test for Local Extrema
Given a critical point \(p\) of a function \(f(x)\):
- if \(f''(p) > 0\), then \(f(x)\) has a local minimum at \(p\).
- if \(f''(p) < 0\), then \(f(x)\) has a local maximum at \(p\).
- if \(f''(p) = 0\), then \(f(x)\) could do anything at \(p\).
Example: The First Derivative Test
Example: Find all local minima and maxima of the function \(f(x) = x^3 - 6x^2 + 9x\). Sketch a graph.
Example: Find all local minima and maxima of the function \(f(x) = x^4 - 8x^2\). Sketch a graph.
Concavity and the Second Derivative
A point \(p\) at which the concavity of \(f(x)\) changes is called an inflection point.
Recall that:
- If \(f''(x) > 0\), then \(f(x)\) is concave up.
- If \(f''(x) < 0\), then \(f(x)\) is concave down.
So to find inflection points, solve \(f''(x) = 0\) and check \(f''(x)\) on either side of the inflection point to see if it changes sign.
Example: Concavity and the Second Derivative
Example: Find all inflection points of the function \(f(x) = x^3 - 6x^2 + 9x\). Sketch a graph.
Example: Find all inflection points of the function \(f(x) = x^4 - 8x^2\). Sketch a graph.
Example: Hard Calculus and Algebra
Example: Sketch a graph of \(f(x) = x^2 \ln(x)\).
