If \(=\mid \begin{array}{ll}\mathrm{x}+\mathrm{a} \sqrt{2} \operatorname{Sin} \mathrm{x}, & 0 \leq \mathrm{x} \leq(\pi / 4) \\ 2 \mathrm{x} \operatorname{Cot} \mathrm{x}+\mathrm{b}, & {[(\pi / 4)<\mathrm{x} \leq(\pi / 2)]} \\ \mathrm{a} \operatorname{Cos} 2 \mathrm{x}+\mathrm{b} \operatorname{Sin} \mathrm{x}, & (\pi / 2)<\mathrm{x} \leq \pi\end{array}\) is continuous on \([0, \pi]\), then \(\mathrm{a}=\ldots \ldots\) and \(\mathrm{b}=\ldots \ldots .\) (a) \(\mathrm{a}=(5 \pi / 2), \mathrm{b}=(5 \pi / 4)\) (b) \(\mathrm{a}=-(5 \pi / 2), \mathrm{b}=-(5 \pi / 4)\) (c) \(\mathrm{a}=(\pi / 6), \mathrm{b}=[(-\pi) / 12]\) (d) \(\mathrm{a}=-(5 \pi / 4), \mathrm{b}=(5 \pi / 2)\)

Step by step solution

01

Analyze the first two parts of the function

At the point \(x = \pi/4\), the first and second parts of the function meet. To ensure continuity, we'll need to make sure that the output of the first and second parts are equal at \(x = \pi/4\): \((x+a\sqrt{2}\sin{x})|_{x=\pi/4} = (2xCot{x}+b)|_{x=\pi/4}\)

02

Calculate the outputs at \(x = \pi/4\)

Now, we'll calculate the outputs of both functions at \(x = \pi/4\): \(\pi/4 + a\sqrt{2}\sin{(\frac{\pi}{4})} = \frac{2(\pi/4)\cot{(\pi/4)}}{b}\)

03

Simplify and solve for a

We can simplify the equation further: \(\pi/4 + a\sqrt{2}\cdot\frac{\sqrt{2}}{2} = \pi/2 + b\) Now, solve for 'a': \(a = \frac{5\pi}{2} - 2b\)

04

Analyze the second and third parts of the function

At the point \(x = \pi/2\), the second and third parts of the function meet. To ensure continuity, we'll need to make sure that the output of the second and third parts are equal at \(x = \pi/2\): \((2xCot{x}+b)|_{x=\pi/2} = (a\cos{2x} + b\sin{x})|_{x=\pi/2}\)

05

Calculate the outputs at \(x = \pi/2\)

Now, we'll calculate the outputs of both functions at \(x = \pi/2\): \((a\cos{\pi} + b\sin{\frac{\pi}{2}}) = (2(\pi/2)\operatorname{Cot}{(\pi/2)}+b)\)

06

Simplify and solve for a

We can simplify the equation further: \(b = -a + b\) Substituting the expression of 'a' from step 3, we get: \(b=(-\frac{5\pi}{2}+2b)\) Now we can solve for 'b': \(b = \frac{5\pi}{4}\) Finally, substituting the value of b found above into the equation from step 1, we obtain the value of a: \(a = \frac{5\pi}{2} - 2 \cdot \frac{5\pi}{4} = \frac{5\pi}{2}\) After finding the values of 'a' and 'b', we can choose the correct answer. The answer is (a) \(\mathrm{a}=(5 \pi / 2), \mathrm{b}=(5 \pi / 4)\).

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