Gilbert Strang’s Calculus: Max, Min, and Second Derivative

Calculus

Derivatives

Second Derivative

Extrema

Author

Chao Ma

Published

February 8, 2026

Video: Max and Min and Second Derivative (Gilbert Strang)

Big idea

The first derivative locates where slope is zero (candidates for max or min). The second derivative tells the curve’s bending direction, so we can decide whether that point is a maximum or minimum.

Speed, slope, and acceleration

Position/height: \(f(t)\) describes where you are.
Speed/slope: \(f'(t)\) is the instantaneous rate of change.
Acceleration/bending: \(f''(t)\) tells how the slope itself changes.

Example: if \(f(x)=x^2\), then \[ f'(x)=2x,\qquad f''(x)=2. \] The constant positive second derivative says the curve bends upward everywhere.

Second derivative for a parabola

Convex, concave, and inflection

If \(f''(x) > 0\), the slope is increasing and the graph bends upward (convex).
If \(f''(x) < 0\), the slope is decreasing and the graph bends downward (concave).
An inflection point is where the bending changes direction, typically where \(f''(x)=0\).

Concave vs. convex and inflection

Example 1: \(y=\sin x\)

\(y' = \cos x\)
\(y'' = -\sin x\)

Sine with slope and curvature

Maximum

At \(x=\pi/2\), \(y=1\) and \(y'=0\). Since \(y''=-1<0\), the curve bends downward, so this is a local maximum.

Inflection point

At \(x=\pi\), \(y''=0\) and the bending changes sign. This is an inflection point.

Minimum

At \(x=3\pi/2\), \(y=-1\) and \(y'=0\). Since \(y''=1>0\), the curve bends upward, so this is a local minimum.

Example 2: \(y=x^3-x^2\)

\(y' = 3x^2-2x = x(3x-2)\)
\(y'' = 6x-2\)

Set \(y'=0\): \[ 3x^2-2x=0 \quad\Rightarrow\quad x=0,\; x=\tfrac{2}{3}. \]

At \(x=0\), \(y''=-2<0\), so this is a local maximum.
At \(x=\tfrac{2}{3}\), \(y''=2>0\), so this is a local minimum.

The inflection point is where \(y''=0\): \[ 6x-2=0 \Rightarrow x=\tfrac{1}{3}. \]

Takeaway. First derivative = where the slope is zero. Second derivative = which way the curve bends. Together they classify maxima, minima, and inflection points.

---
title: "Gilbert Strang's Calculus: Max, Min, and Second Derivative"
author: "Chao Ma"
date: "2026-02-08"
categories: [Calculus, Derivatives, Second Derivative, Extrema]
---

Video: [Max and Min and Second Derivative (Gilbert Strang)](https://www.youtube.com/watch?v=tBBJ2TSTa1Q&t=974s)

{{< video https://www.youtube.com/watch?v=tBBJ2TSTa1Q&t=974s >}}

# Big idea

The **first derivative** locates where slope is zero (candidates for max or min). The **second derivative** tells the curve's bending direction, so we can decide whether that point is a maximum or minimum.

# Speed, slope, and acceleration

- **Position/height**: $f(t)$ describes where you are.
- **Speed/slope**: $f'(t)$ is the instantaneous rate of change.
- **Acceleration/bending**: $f''(t)$ tells how the slope itself changes.

Example: if $f(x)=x^2$, then
$$
f'(x)=2x,\qquad f''(x)=2.
$$
The constant positive second derivative says the curve bends upward everywhere.

![Second derivative for a parabola](max-min-second-derivative-1.png)

# Convex, concave, and inflection

- If $f''(x) > 0$, the slope is increasing and the graph bends **upward** (convex).
- If $f''(x) < 0$, the slope is decreasing and the graph bends **downward** (concave).
- An **inflection point** is where the bending changes direction, typically where $f''(x)=0$.

![Concave vs. convex and inflection](max-min-second-derivative-2.png)

# Example 1: $y=\sin x$

- $y' = \cos x$
- $y'' = -\sin x$

![Sine with slope and curvature](max-min-second-derivative-3.png)

## Maximum

At $x=\pi/2$, $y=1$ and $y'=0$. Since $y''=-1<0$, the curve bends downward, so this is a **local maximum**.

## Inflection point

At $x=\pi$, $y''=0$ and the bending changes sign. This is an **inflection point**.

## Minimum

At $x=3\pi/2$, $y=-1$ and $y'=0$. Since $y''=1>0$, the curve bends upward, so this is a **local minimum**.

# Example 2: $y=x^3-x^2$

- $y' = 3x^2-2x = x(3x-2)$
- $y'' = 6x-2$

Set $y'=0$:
$$
3x^2-2x=0 \quad\Rightarrow\quad x=0,\; x=\tfrac{2}{3}.
$$

- At $x=0$, $y''=-2<0$, so this is a **local maximum**.
- At $x=\tfrac{2}{3}$, $y''=2>0$, so this is a **local minimum**.

The inflection point is where $y''=0$:
$$
6x-2=0 \Rightarrow x=\tfrac{1}{3}.
$$

---

**Takeaway.** First derivative = where the slope is zero. Second derivative = which way the curve bends. Together they classify maxima, minima, and inflection points.