Training Optimization Unconstrained Optimization
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Unconstrained Optimization

24 min Optimization

Unconstrained Optimization

Critical Points (Single Variable)

Find $x$ where $f'(x) = 0$ or $f'(x)$ is undefined. Test with the second derivative:

  • $f''(x) > 0$: local minimum
  • $f''(x) < 0$: local maximum
  • $f''(x) = 0$: inconclusive — use first-derivative test
Closed Interval Method

To find the absolute max/min of $f$ on $[a,b]$:

  1. Find all critical points in $(a,b)$
  2. Evaluate $f$ at each critical point and at $a$, $b$
  3. The largest value is the absolute max; smallest is the absolute min
Extreme Value Theorem

If $f$ is continuous on $[a,b]$, then $f$ attains an absolute maximum and minimum on $[a,b]$.

Example 1

Find the absolute extrema of $f(x) = x^3 - 3x + 1$ on $[-2, 2]$.

$f'(x) = 3x^2 - 3 = 0 \Rightarrow x = \pm 1$.

$f(-2) = -1$, $f(-1) = 3$, $f(1) = -1$, $f(2) = 3$.

Absolute max $= 3$ at $x=-1$ and $x=2$; absolute min $= -1$ at $x=-2$ and $x=1$.

Example 2

Minimize $f(x) = x + 4/x$ for $x > 0$.

$f'(x) = 1 - 4/x^2 = 0 \Rightarrow x = 2$. $f''(2) = 8/8 = 1 > 0$: minimum.

$f(2) = 2 + 2 = 4$.

Example 3

Find the inflection point of $f(x) = x^3$.

$f''(x) = 6x = 0$ at $x = 0$. Concavity changes sign → inflection at $(0, 0)$.

Practice Problems

1. Critical points of $f(x) = x^4 - 4x^2$.
2. Classify each critical point in Problem 1.
3. Absolute max/min of $f(x) = x^2 - 4x + 5$ on $[0, 4]$.
4. Minimize $f(x) = 2x + 18/x$, $x > 0$.
5. Inflection points of $f(x) = x^4 - 6x^2$.
6. Extrema of $f(x) = e^{-x^2}$.
7. What does the first-derivative test say when $f'$ changes from $+$ to $-$?
8. Find the global min of $f(x) = |x-3| + 1$.
9. Critical points of $f(x) = \sin x$ on $[0, 2\pi]$.
10. Max of $f(x) = -x^2 + 6x - 5$.
11. Why check endpoints on $[a,b]$?
12. $f(x) = x^{2/3}$: critical point? Classify.
Show Answer Key

1. $f'=4x^3-8x=4x(x^2-2)=0$; $x=0,\pm\sqrt{2}$

2. $f''=12x^2-8$. At $x=0$: $f''=-8<0$ (max). At $x=\pm\sqrt{2}$: $f''=16>0$ (min).

3. $f'=2x-4=0$ at $x=2$. $f(0)=5$, $f(2)=1$, $f(4)=5$. Min $=1$, max $=5$.

4. $f'=2-18/x^2=0$; $x=3$. $f(3)=12$.

5. $f''=12x^2-12=0$; $x=\pm 1$

6. $f'=-2xe^{-x^2}=0$ at $x=0$. $f(0)=1$; global max (function → 0 as $|x|\to\infty$).

7. Local maximum

8. At $x=3$: $f(3)=1$ (global min)

9. $f'=\cos x=0$ at $x=\pi/2$ (max), $x=3\pi/2$ (min)

10. $f'=-2x+6=0$; $x=3$. $f(3)=4$.

11. Absolute extrema can occur at endpoints, not just critical points

12. $f'=\frac{2}{3}x^{-1/3}$, undefined at $x=0$. Cusp — minimum $f(0)=0$.

Applied Max-Min Problems