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Chapter 9: Problem 32

Find the max or min values of \(f(x)=2 \sin x+\cos 2 x\)

Short Answer

Expert verified
The specific critical points, maxima and minima values depend on the solution to the equation \(\cos x = \sin 2x\). Once these values are found, the maxima and minima will repeat every \(2\pi\) interval due to the periodic nature of the functions involved.

Step by step solution

01

Derive the Function

The student needs to calculate the derivative of the function \(f(x)\). The derivative of \(f(x)\) is \(f'(x)=2 \cos x - 2 \sin 2x\).
02

Find the Critical Points

The critical points of the function are found by setting the derivative equal to zero, that is, solving \(2 \cos x - 2 \sin 2x = 0\). This gives \(2 \cos x = 2 \sin 2x\), and then \( \cos x = \sin 2x = 0\). Use the properties of the sine and cosine functions to find all solutions in the interval \([0, 2\pi]\).
03

Classify the Critical Points

To classify the critical points, compute the second derivative which is \(f''(x)=-2\sin x-4\cos 2x\). Use the second derivative test to classify each critical point as a local minimum, local maximum, or neither.
04

Calculate Maxima and Minima

The local maxims and minimums are found by evaluating the original function \(f(x)=2 \sin x+\cos 2 x\) at the critical points and endpoints (if applicable). Keep in mind that these are periodic functions, so all solutions are valid for \(x = x+2n\pi\) where \(n\) is an integer, and the maximum and minimum points will repeat every \(2\pi\) interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative of Trigonometric Functions
Calculating the derivative of a function allows us to determine how the function changes. For trigonometric functions, derivatives play a crucial role in finding these rates of change.
For our function, we have found the derivative to be \(f'(x) = 2 \cos x - 2 \sin 2x\), and the process involves knowing some key derivative rules:
  • The derivative of \(\sin x\) is \(\cos x\).
  • The derivative of \(\cos x\) is \(-\sin x\).
  • The derivative of \(\sin 2x\), involving a chain rule, is \(2 \cos 2x\).
These derivatives help us determine where the function increases or decreases. Understanding these principles is essential for advancing further into optimization problems.
This derivative essentially maps the slope of our function at any given point \(x\), providing insights into the behavior of the function's graph.
Critical Points
Critical points occur where the derivative of the function equals zero or is undefined. These points are crucial since they offer potential locations for maxima and minima.
To find them, we solve the equation \(2 \cos x - 2 \sin 2x = 0\). This simplifies to \(\cos x = \sin 2x\) in this context. Solving this equation within an interval, such as \([0, 2\pi]\), will reveal points where the function might change direction.

These critical points mark significant transitions in a function's behavior, often indicating peaks or troughs. By determining the sign changes around these points, we can further analyze the function's graph and behavior.
Second Derivative Test
The second derivative test helps to classify critical points further. By computing the second derivative, we gain insight into the concavity of the function. For our example, the second derivative is \(f''(x) = -2\sin x - 4\cos 2x\).
  • If \(f''(x) > 0\) at a point, the function is concave up, indicating a local minimum.
  • If \(f''(x) < 0\), it means concave down, suggesting a local maximum.
  • If \(f''(x) = 0\), the test is inconclusive.
Using these signs allows us to determine which critical points are maximums, minimums, or neither.
Applying this method to each critical point provides more detailed insights into the function's overall shape and helps identify specific roles of each point.
Periodic Functions
Trigonometric functions are often periodic, meaning they repeat their values in regular intervals. For sine and cosine functions, this interval is \(2\pi\).
When optimizing these functions, this periodicity means any local maxima or minima found in one period \([0, 2\pi]\) will repeat at intervals of \(x + 2n\pi\) for any integer \(n\).
It's crucial to consider this repetition property when analyzing or applying trigonometric functions in real-world scenarios, as it simplifies understanding how values might evolve beyond the initial phase.
Recognizing the periodic nature allows for easier predictions and understandings of phenomena described by these mathematical functions.

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Most popular questions from this chapter

The point \((0,5)\) is closest to the curve \(x^{2}=2 y\) is (a) \((2 \sqrt{2}, 0)\) (b) \((0,0)\) (c) \((2,2)\) (d) None

The difference between the greatest and the least values of the function \(f(x)=\int_{0}^{x}\left(a t^{2}+1+\operatorname{cost}\right) d t, a>0 x \in[2,3]\) is (a) \(\frac{19}{3} a+1+\sin 3-\sin 2\) (b) \(\frac{18}{3} a+1+2 \sin 3\) (c) \(\frac{18}{3} a-1+2 \sin 3\) (d) None

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