Problem 69 - Algebra and Trigonometry

Question

\(67-70\). Find the maximum and minimum values of the function. $$ y=2 \sin x+\sin ^{2} x $$

Step-by-Step Solution

Answer

The maximum value is 3 and the minimum value is -1.

Step 1: Identify the Function

The function given is \( y = 2 \sin x + \sin^2 x \). To find its maximum and minimum values, we need to understand how it behaves with respect to \( x \).

Step 2: Differentiate the Function

To find the critical points, we differentiate the function: \( \frac{dy}{dx} = 2\cos x + 2\sin x \cdot \cos x \). This can be simplified to \( \frac{dy}{dx} = 2\cos x (1 + \sin x) \).

Step 3: Find Critical Points

Set \( \frac{dy}{dx} = 0 \):\[ 2\cos x (1 + \sin x) = 0 \]This gives solutions \( \cos x = 0 \) and \( \sin x = -1 \). For \( \cos x = 0 \), \( x = \frac{\pi}{2} + n\pi \). For \( \sin x = -1 \), \( x = \frac{3\pi}{2} + 2n\pi \).

Step 4: Evaluate the Function at Critical Points

Evaluate \( y \) at the critical points:- For \( x = \frac{\pi}{2} + n\pi \), \( \sin x = \pm 1 \): - If \( \sin x = 1 \), \( y = 2(1) + 1^2 = 3 \) - If \( \sin x = -1 \), \( y = 2(-1) + (-1)^2 = -1 \)

Step 5: Check End Behavior and Other Values

Consider the interval [0, 2\(\pi\)] to find any possible additional considerations for maxima or minima over one full period of the sine function. The values at the boundary for sine are not maximum or minimum outside the periodic solutions already found.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative provides information about the rate at which the function's value changes with respect to changes in its input value, typically noted as \(x\). In this exercise, we are given a function:

• \( y = 2 \sin x + \sin^2 x \)

To find its critical points, we differentiate it with respect to \(x\). The derivative gives us the slope of the tangent line to the curve at any point, which is crucial for identifying where the slope is zero, indicating potential extremes (maximum or minimum values).

Using standard differentiation rules and trigonometric identities, we compute the derivative:

• \( \frac{dy}{dx} = 2 \cos x + 2 \sin x \cdot \cos x = 2 \cos x (1 + \sin x) \)

This derivative helps us locate where changes in the original function's behavior occur, setting the stage for finding critical points by solving \( \frac{dy}{dx} = 0 \). Differentiation thus transforms static equations into dynamic tools for exploring functions.

Critical Points

Critical points are values of \(x\) at which the derivative of a function is zero or undefined. They represent potential locations of maximum, minimum, or inflection points, where the function's behavior changes. In our function \( y = 2 \sin x + \sin^2 x \), once we have the derivative \( \frac{dy}{dx} = 2 \cos x (1 + \sin x) \), we can determine its critical points by solving:

• \( 2 \cos x (1 + \sin x) = 0 \)

This equation implies two scenarios:

• \( \cos x = 0 \)
• \( 1 + \sin x = 0 \Rightarrow \sin x = -1 \)

For \( \cos x = 0 \), \( x = \frac{\pi}{2} + n\pi \), and for \( \sin x = -1 \), \( x = \frac{3\pi}{2} + 2n\pi \) where \(n\) is an integer.

These critical points give specific \(x\) values where the function might experience extremes, crucial for the next step in determining if these points are indeed maximum or minimum values.

Maximum and Minimum Values

Maximum and minimum values of a function are the greatest and smallest y-values (outputs) that the function can achieve. Once we have identified the critical points, evaluating the function at these points helps deduce whether they correspond to maxima or minima. For our function:

• For \( x = \frac{\pi}{2} + n\pi \):
• \( \sin x = 1 \Rightarrow y = 3 \)
• \( \sin x = -1 \Rightarrow y = -1 \)
• For \( x = \frac{3\pi}{2} + 2n\pi \), \( \sin x = -1 \Rightarrow y = -1 \)

Thus, the maximum value of the function is 3, and the minimum value is -1 within one full period \( \[0, 2\pi\] \) of sine.

These values indicate the peaks and valleys of our function over its cycle, showing where it reaches its highest and lowest points, which are essential in many areas of mathematical analysis and real-world applications.

Sine Function

The sine function is a primary trigonometric function dealing with angles and oscillatory phenomena, such as waves. It is periodic with a cycle length of \(2\pi\), meaning every \(2\pi\) units, it repeats its pattern. Its range is between -1 and 1, influencing how functions built around sine behave. In the function:

• \( y = 2 \sin x + \sin^2 x \)

The sine component \( \sin x \) affects the shape of the curve as \(x\) varies. The squared sine \( \sin^2 x \) modulates this curve further, adding variety and influencing where peaks and valleys can occur.

Considering its range and periodicity, finding critical points and extreme values requires understanding sine's repeating nature. This approach allows functions like ours to determine outcomes reliably within cycles, ensuring periodic challenges are resolved in trigonometric exercises. The ability to effectively use properties of the sine function is vital for solving mathematical problems involving periodic motion and oscillations.