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Chapter 4: Problem 14

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=\sin x-\cos x ;[0, \pi]$$

Short Answer

Expert verified
Maximum: \(\sqrt{2}\) at \(x=\frac{3\pi}{4}\); Minimum: \(-1\) at \(x=0\).

Step by step solution

01

Identify Critical Points

To find critical points, we first differentiate the function \(f(x) = \sin x - \cos x\). The derivative is \(f'(x) = \cos x + \sin x\). Set the derivative equal to zero to find critical points: \(\cos x + \sin x = 0\). Solve for \(x\):\[\tan x = -1 \\Rightarrow x = \frac{3\pi}{4}\] since it must be within \([0, \pi]\).
02

Evaluate Function at Critical Points

Evaluate \(f(x)\) at the critical point and at the endpoints of the interval. Compute \(f(0), f(\pi),\) and \(f\left(\frac{3\pi}{4}\right)\):\[f(0) = \sin(0) - \cos(0) = 0 - 1 = -1 \f(\pi) = \sin(\pi) - \cos(\pi) = 0 + 1 = 1 \f\left(\frac{3\pi}{4}\right) = \sin\left(\frac{3\pi}{4}\right) - \cos\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} - \left(-\frac{\sqrt{2}}{2}\right) = \sqrt{2} \approx 1.414\]
03

Determine the Absolute Maximum and Minimum

Compare the function values from Step 2. The values are:- \(f(0) = -1\)- \(f(\pi) = 1\)- \(f\left(\frac{3\pi}{4}\right) = \sqrt{2}\)The absolute maximum value is \(f\left(\frac{3\pi}{4}\right) = \sqrt{2}\) (approximately 1.414), and it occurs at \(x = \frac{3\pi}{4}\). The absolute minimum value is \(f(0) = -1\), occurring at \(x = 0\).

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Key Concepts

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

Critical Points
In calculus, critical points are crucial in finding the extremum points of a differentiable function. To locate these points for a function like \(f(x) = \sin x - \cos x\), you'll need to find its derivative and set it to zero. The derivative here is \(f'(x) = \cos x + \sin x\). By solving the equation \(\cos x + \sin x = 0\), you identify where the slope of the tangent to the curve is zero, indicating a potential extremum.
This process resulted in the critical point \(x = \frac{3\pi}{4}\). Remember, finding critical points requires checking the range you're investigating, as not all solutions might be within your interval.
Absolute Maximum and Minimum
Absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function achieves over that specific range. For \(f(x) = \sin x - \cos x\) on \([0, \pi]\), you begin by evaluating \(f\) at critical points and endpoints of the interval.
  • At \(x = 0\), \(f(0) = -1\).
  • At \(x = \frac{3\pi}{4}\), \(f\left(\frac{3\pi}{4}\right) \approx 1.414\).
  • At \(x = \pi\), \(f(\pi) = 1\).
Comparing these results, \(f\left(\frac{3\pi}{4}\right)\) is the absolute maximum because it is the highest value, and \(f(0)\) is the absolute minimum as it is the lowest.
Derivative
The derivative is a fundamental concept that describes the rate at which a function changes. For this problem, the derivative of \(f(x) = \sin x - \cos x\) is \(f'(x) = \cos x + \sin x\).
  • The derivative was used to find critical points by setting \(f'(x) = 0\).
  • This step ensures that the slope is zero, indicating a possible extremum like a maximum or minimum.
Understanding how derivatives help determine the shape and behavior of a function is essential for solving optimization problems like this.
Trigonometric Functions
Trigonometric functions play a pivotal role in calculus, especially when dealing with periodic phenomena. This exercise involved the sine and cosine functions in the function \(f(x) = \sin x - \cos x\).
  • These functions are differentiable over their entire domains, allowing the application of calculus tools like derivatives.
  • The critical point found amidst trigonometric functions \(x = \frac{3\pi}{4}\) aligns with specific angles where ratios, such as \(\tan x = -1\), denote special relationships between sine and cosine.
Trigonometric identities and their derivatives are invaluable for analyzing and comprehending periodic functions effectively.

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Most popular questions from this chapter

At a construction site, a worker raises an 8 -foot sheet of plywood to a platform by sliding the sheet up over the edge of the platform in such a way that the horizontal distance between the bottom of the plywood and the platform edge is kept at 1 foot. If the distance along the plywood from its bottom to the platform edge measures \(x\) feet, then the top of the plywood will be $$y=\left(\frac{8}{x}-1\right) \sqrt{x^{2}-1} \text { feet }$$ above the platform \((1 \leq x \leq 8) .\) What is the greatest distance above the platform that the top of the plywood reaches?

Suppose that \(x=x_{0}\) is a point at which a function \(f\) is continuous but not differentiable and that \(f^{\prime}(x)\) approaches different finite limits as \(x\) approaches \(x_{0}\) from either side. Invent your own term to describe the graph of \(f\) at such a point and discuss the appropriateness of your term.

(a) Determine whether the following limits exist, and if so, find them: $$\lim _{x \rightarrow+\infty} e^{x} \cos x, \quad \lim _{x \rightarrow-\infty} e^{x} \cos x$$ (b) Sketch the graphs of the equations \(y=e^{x}, y=-e^{x}\) and \(y=e^{x} \cos x\) in the same coordinate system. and label any points of intersection. (c) Use a graphing utility to generate some members of the family \(y=e^{a x} \cos b x(a>0 \text { and } b>0),\) and discuss the cffect of varying \(a\) and \(b\) on the shape of the curve.

Determine whether the statements are true or false If a statement is false, find functions for which the statement fails to hold. (a) If \(f\) and \(g\) are concave up on an interval, then so is \(f+g\) (b) If \(f\) and \(g\) are concave up on an interval, then so is \(f \cdot g\)

Let \(f(x)=a x^{2}+b x+c,\) where \(a>0 .\) Prove that \(f(x) \geq 0\) for all \(x\) if and only if \(b^{2}-4 a c \leq 0 .\) [Hint: Find the minimum of \(f(x) .]\)
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