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Chapter 4: Problem 49
Show that the equation \(x^{5}+3 x^{4}+x-2=0\) has atleast one root in \([0,1]\).
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Let \(f\) be a continuous function on \(R\) such that \(f\left(\frac{1}{4^{n}}\right)=\left(\sin e^{n}\right) e^{-n^{2}}+\frac{n^{2}}{n^{2}+1}\), then find \(f(0)\).
If \(f(x)= \begin{cases}\frac{\sin (a+1) x+\sin x}{x} & : x<0 \\ c & : x=0 \text { is continuous } \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}} & : x>0\end{cases}\) at \(x=0\), then (a) \(a=-3 / 2, b=0, c=1 / 2\) (b) \(a=-3 / 2, b=1, c=-1 / 2\) (c) \(a=-3 / 2, b=R, c=1 / 2\) (d) None
Show that the equation \(x^{3}-3 x+1=0\) has a real root in \([1,2]\).
Check the differentiability of the function \(f(x)=\left|x^{2}-1\right|+\left|x^{2}-4\right|\) in \(R\).
Let \(f(x)= \begin{cases}\frac{x e^{\frac{1}{x}}}{1+e^{\frac{1}{2}}} & : x \neq 0 \\ 0 & : x=0\end{cases}\) Examine the continuity and the differentiability at \(x=0 .\)
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