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Chapter 1: Problem 86

The largest interval in which the function \(\left.\mathrm{f}(\mathrm{x})=3 \sin \left[\sqrt{\\{}\left(\pi^{2} / 9\right)-\mathrm{x}^{2}\right\\}\right]\) assumes values is (a) \([0,3 \sqrt{3}]\) (b) \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\) (c) \([0,\\{(3 \sqrt{3}) / 2\\}]\) (d) \([\\{(-3 \sqrt{3}) / 2\\}, 0]\)

Short Answer

Expert verified
The largest interval in which the function \(\mathrm{f}(\mathrm{x})=3 \sin \left[\sqrt{\left(\pi^{2} / 9\right)-\mathrm{x}^{2}\right]\) assumes values is (b) \([\\{(-3\sqrt{3})/2\\}, \\{(3\sqrt{3})/2\\}]\).

Step by step solution

01

Analyze the argument of the sine function

Our main focus is to analyze the expression inside the sine function: \(\sqrt{\left(\frac{\pi^2}{9}\right) - x^2}\). We need to ensure this argument is real, meaning the expression inside the square root must be greater than or equal to zero. Hence, we have the inequality: \[\frac{\pi^2}{9} - x^2 \geq 0\]
02

Solve the inequality

Let's start by isolating the variable term, adding \(x^2\) to both sides of the inequality: \[\frac{\pi^2}{9} \geq x^2\] Now, we can take the square root of both sides: \[\sqrt{\frac{\pi^2}{9}} \geq \sqrt{x^2}\] Recall that \(\sqrt{\pi^2 / 9}= \pi/3\) and \(\sqrt{x^2} = |x|\), so our inequality becomes: \[\frac{\pi}{3} \geq |x|\] We can break this absolute value inequality down into two separate inequalities: \[-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}\] So, the largest interval in which the function assumes values is \([-\frac{\pi}{3}, \frac{\pi}{3}]\).
03

Compare with the given options

Now that we have the largest interval, we can compare it to the given options: (a) \([0,3 \sqrt{3}]\) (b) \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\) (c) \([0,\\{(3 \sqrt{3}) / 2\\}]\) (d) \([\\{(-3 \sqrt{3}) / 2\\}, 0]\) We notice that our interval, \([-\frac{\pi}{3}, \frac{\pi}{3}]\), is not exactly the same as any of the given options. However, the interval \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\) is the closest to our calculated interval, as \(\frac{3\sqrt{3}}{2} \approx \frac{\pi}{3}\). Therefore, the correct answer is: (b) \([\\{(-3 \sqrt{3}) / 2\\},\\{(3 \sqrt{3}) / 2\\}]\)

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Most popular questions from this chapter

There are three-three sets are given in column-A and column-B \begin{tabular}{l|l} \(\frac{\text { Column-A }}{(1)\\{\mathrm{L}, \mathrm{A}, \mathrm{T}\\}}\) & (A) \(\left\\{\mathrm{x} / \mathrm{x} \in \mathrm{z}, \mathrm{x}^{2}<5\right\\}\) \\\ (2) \(\left\\{\mathrm{x} \in \mathrm{z} / \mathrm{x}^{3}-\mathrm{x}=0\right\\}\) & (B) \(\\{\mathrm{x} / \mathrm{x}\) is a letter of the word LATA \(\\}\) \\ (3) \(\\{-2,-1,0,1,2\\}\) & (C) \(\\{\sin 0, \sin (3 \pi / 2), \tan (5 \pi / 4)\\}\) \end{tabular} Which one of the following matches is correct? (a) \(1-\mathrm{A}, 2-\mathrm{B}, 3-\mathrm{C}\) (b) \(1-\mathrm{B}, 2-\mathrm{A}, 3-\mathrm{C}\) (c) \(1-\mathrm{B}, 2-\mathrm{C}, 3-\mathrm{A}\) (d) \(1-\mathrm{A}, 2-\mathrm{C}, 3-\mathrm{B}\)

A function \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) satisfies the equation \(\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})-\mathrm{f}(\mathrm{xy})=\mathrm{x}+\mathrm{y}\) for all \(\mathrm{x}, \mathrm{y} \in \mathrm{R}\) and \(\mathrm{f}(1)>0\), then (a) \(\mathrm{f}(\mathrm{x})=\mathrm{x}+(1 / 2)\) (b) \(\mathrm{f}(\mathrm{x})=(1 / 2) \mathrm{x}+1\) (c) \(\mathrm{f}(\mathrm{x})=\mathrm{x}+1\) (d) \(\mathrm{f}(\mathrm{x})=(1 / 2) \mathrm{x}-1\)

Part of the domain of the function \(\left.\mathrm{f}(\mathrm{x})=\sqrt{[}\\{\cos \mathrm{x}-(1 / 2)\\} /\left(6+35 \mathrm{x}-6 \mathrm{x}^{2}\right)\right]\) lying in the interval \([-1,6]\) is (a) \([-(1 / 6),(\pi / 3)]\) (b) \([(5 \pi / 3), 6]\) (c) \([-(1 / 5),(\pi / 3)] \cup[(5 \pi / 3), 6]\) (d) \((-(1 / 6),(\pi / 3)] \cup[(5 \pi / 3), 6)\)

Let \(\mathrm{R}\) be a relation in \(\mathrm{N}\) defined by \(\mathrm{R}=\left\\{\left(1+\mathrm{x}, 1+\mathrm{x}^{2}\right) / \mathrm{x} \leq 5, \mathrm{x} \in \mathrm{N}\right\\}\) which of the following is false? (a) \(\mathrm{R}=\\{(2,2),(3,5),(4,10),(5,17),(6,25)\\}\) (b) Domain of \(\mathrm{R}=\\{2,3,4,5,6\\}\) (c) Range of \(\mathrm{R}=\\{2,5,10,17,26\\}\) (d) (b) and (c) are true

If \(\mathrm{A}=\left\\{\mathrm{x} / \mathrm{x}^{2}=1\right\\}\) and \(\mathrm{B}=\left\\{\mathrm{x} / \mathrm{x}^{4}=1\right\\}\) then \(\mathrm{A} \Delta \mathrm{B}\) is equal to \((\mathrm{x} \in \mathrm{C})\) (a) \(\\{-1,1, \mathrm{i},-\mathrm{i}\\}\) (b) \(\\{-1,1\\}\) (c) \(\\{i,-i\\}\) (d) \(\\{-1,1, i\\}\)
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