# Maxima and Minima with Trigonometric Functions

A trigonometric equation can model periodic motions. By differentiating these functions, we can solve problems relating to maxima (maximums) and minima (minimums).
Remember that the following steps are used when solving a maximum or minimum problem.

• Step 1: Find $f^{\prime}(x)$ to obtain the gratest function.
• Step 2: Solve for $x$ where $f'(x)=0$ to find the values of $x$ where the maximum or minimum occur.
• Step 3: Apply the first derivative test to see if the point is a maximum or minimum.
• Step 4: Substitute the appropriate value of $x$ into $f(x)$ to obtain the maximum or minimum.

### Example 1

The population of a colony of ants rises and falls according to the breeding season. The population can be modelled by the equation $P(t) = 100\sin{\dfrac{\pi t}{4}}+4000$, where $t$ is the number of months since the beginning of the year.

(a)    Find the first time at which the population is greatest.

\begin{align} \displaystyle P^{\prime}(t) &= 100 \times \dfrac{\pi}{4} \cos{\dfrac{\pi t}{4}} \\ 100 \times \dfrac{\pi}{4} \cos{\dfrac{\pi t}{4}} &= 0 \\ \dfrac{25\pi}{4} \cos{\dfrac{\pi t}{4}} &= 0 \\ \cos{\dfrac{\pi t}{4}} &= 0 \\ \dfrac{\pi t}{4} &= \dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{5\pi}{2}, \cdots \\ t &= 2, 6, 10, \cdots \end{align}
The maximum population first occurs after 2 months.

(b)   Find the maximum population.

\begin{align} \displaystyle P(2) &= 100\sin{\dfrac{\pi \times 2}{4}}+4000 \\ &= 100\sin{\dfrac{\pi}{2}}+4000 \\ &= 100 \times 1+4000 \\ &= 4100 \end{align}
The maximum population is $4100$.

Note:
Finding the derivatives of the functions is not always the case. Sometimes, you can find cases where you do not have to differentiate the functions, but using the given domain and associated ranges will give better solutions. Let’s take a look at the following Example 2 for reference.

### Example 2

Find the maximum and minimum values of $f(x)=2\cos{\dfrac{\pi t}{2}+5}+1$.

\begin{align} \displaystyle -1 &\le \cos{t} \le 1 \\ -1 &\le \cos{\dfrac{\pi t}{2}} \le 1 \\ 0 &\le \cos{\dfrac{\pi t}{2}+1} \le 2 \\ 0 &\le 2\cos{\dfrac{\pi t}{2}+1} \le 4 \end{align}
Therefore the maximum value is $4$, and the minimum value is $0$.

Discover more enlightening videos by visiting our YouTube channel!

## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…

## Simplified Calculus: Exploring Differentiation by First Principles

Exploring differentiation by first principles Suppose we are given a function $f(x)$ and asked to find its derivative at the point where $x=a$. This is…

## Induction Made Simple: The Ultimate Guide

“Induction Made Simple: The Ultimate Guide” is your gateway to mastering the art of mathematical induction, demystifying a powerful tool in mathematics. This ultimate guide…

## High School Math for Life: Making Sense of Earnings

Salary Salary refers to the fixed amount of money that an employer pays an employee at regular intervals, typically on a monthly or biweekly basis,…

## Quick Guide to Max Range in Inclined Plane Motion

Projectile motion on an inclined plane is a fascinating variation of the classic projectile motion problem. In this scenario, the trajectory of the projectile is…