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Chapter 5: Problem 7
Let \(f(x)=\frac{\sin x}{x}\), where \(0
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The set of all values of ' \(a\) ' for which the function, \(f(x)=\left(a^{2}-3 a+2\right)\left(\cos ^{2} \frac{x}{4}-\sin ^{2} \frac{x}{4}\right)+(a-1) x+\) sin 1 does not possess critical points is: (a) \([1, \infty)\) (b) \((0,1) \cup(1,4)\) (c) \((-2,4)\) (d) \((1,3) \cup(3,5)\)
At \(x \rightarrow 0^{+}\), all of these function \(\frac{1}{x} \frac{1}{x^{2}}, \frac{1}{\sqrt{x}}\) become infinite. Which of these increases most rapidly: (a) \(\frac{1}{x}\) (b) \(\frac{1}{x^{2}}\) (c) \(\frac{1}{\sqrt{x}}\) (d) all increase with equal rate
In which of the following functions Rolle's Theorem is applicable? (a) \(f(x)=\left\\{\begin{array}{ll}x, & 0 \leq x<1 \\ 0, & x=1\end{array}\right.\) on \([0,1]\) (b) \(f(x)=\left\\{\begin{array}{ll}\frac{\sin x}{x}, & -\pi \leq x<0 \\ 0, & x=0\end{array}\right.\) on \([-\pi, 0]\) (c) \(f(x)=\frac{x^{2}-x-6}{x-1}\) on \([-2,3]\) (d) \(f(x)=\left\\{\begin{array}{cc}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1, \text { on }[-2,3] \\ -6 & \text { if } x=1\end{array}\right.\)
The minimum value of \(\left(1+\frac{1}{\sin ^{n} \alpha}\right)\left(1+\frac{1}{\cos ^{n} \alpha}\right)\) for \(\alpha \in\left(0, \frac{\pi}{2}\right)\) is given by (a) 1 (b) 2 (c) \(\left(1+2^{m 2}\right)^{2}\) (d) None of these
Let \(f(x)=2+\cos x\) for all real \(x\) A: For each real \(t\), there exists a point ' \(c^{\prime}\) in \([\mathrm{t}, t+\pi)\) such that \(\mathrm{f}^{\prime}(\mathrm{c})=0\) R: \(f(t)=f(t+2 \pi)\) for each real \(t\).
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