11.5 Maximum and Minimum Values

11.5 Maximum and Minimum Values

For $f(\theta) = a + b\sin(c\theta + d)$ or $a + b\cos(c\theta + d)$:

• Vertical shift: $a$
• Amplitude: $|b|$
• Period: $\dfrac{2\pi}{|c|}$
• Phase shift from $d$

Since $|\sin| \le 1$ and $|\cos| \le 1$:

Also: amplitude $= \dfrac{M - m}{2}$.

For a reciprocal $1/g(\theta)$ (when $g$ keeps constant sign), max of $1/g$ is $1/m$ and min is $1/M$.

Example 2

(i) $2 + 3\sin x$: $M = 5$, $m = -1$.

(ii) $5 - 2\cos 3x$: $M = 7$, $m = 3$.

(iii) $\dfrac{1}{5-2\cos 3x}$: max $\dfrac{1}{3}$, min $\dfrac{1}{7}$.