Chapter 3

If \(f(x)\) and \(g(x)\) are two functions continuous everywhere and \(h(x)=\lim _{n \rightarrow \infty} \frac{f(x)+x^{2 n} g(x)}{1+x^{2 n}}\), then prove that \(h(x)\) is continuous everywhere except at \(x=1,-1\). Find the condition on \(f(x)\) and \(g(x)\) which makes \(h(x)\) continuous everywhere.

If \(f(x+2 y)=f(x)[f(y)]^{2} \forall x, y\) and \(f(x)\) is continuous at \(x=0\), then check the continuity of \(f(x)\).

Can one assert that the square of a discontinuous function is also a discontinuous function? Give an example of a function discontinuous everywhere whose square is a continuous function.

Let \(f(x)\) be a continuous and \(g(x)\) be a discontinuous function. Prove that \(f(x)+g(x)\) is a discontinuous function.

Does the function \(f(x)=\frac{x^{3}}{4}-\sin \pi x+3\) take the value \(2 \frac{1}{3}\) within the interval \([-2,2]\) ?

Given \(f(x)=x^{3}+x+1\), show that \(f(x)\) has a zero in the interval \([-1,0]\).

Show that the equation \(x^{5}-3 x-1=0\) has at least one root lying between 1 and 2 .

Show that the equation \(x 2^{x}=1\) has at least one positive root not exceeding 1 .

Show that \(f(x)=x^{3}-3 x+1\) has a zero in the interval \([1,2]\).

Does the equation \(x^{5}-18 x+2=0\) has a root in the interval \([-1,1]\) ?

Does the equation \(\sin x-x+1=0\) has a root?

Show that the equation \(x=a \sin x+b\), where \(00\), has at least one positive root which does not exceed \(a+b\).

Show that the equation \(x=a \sin x+b\), where \(00\), has at least one positive root which does not exceed \(a+b\).

Show that \(x+\ln x=0\) has a solution in the interval \((0,1)\).

Show that the equation \(\frac{a_{1}}{x-\lambda_{1}}+\frac{a_{2}}{x-\lambda_{2}}+\frac{a_{3}}{x-\lambda_{3}}=0\), where \(a_{1}>0, a_{2}>0, a_{3}>0\) and \(\lambda_{1}<\lambda_{2}<\lambda_{3}\), has two real roots lying in the intervals \(\left(\lambda_{1}, \lambda_{2}\right)\) and \(\left(\lambda_{2}, \lambda_{3}\right)\).

Given \(\begin{aligned} f(x) &=x^{2}+1, &-2 \leq x<0 \\ &=-x^{2}-1, & 0 & \leq x \leq 2 . \end{aligned}\) Is there a point in the interval \([-2,2]\) at which \(f(x)=0 ?\)

The function \(f(x)\) is continuous in the interval \([a, b]\) and has values of the same sign on its end-points. Can one assert that there is no point in \([a, b]\) at which the function becomes zero?

Show that the function \(f(x)=x^{5}-4 x+1\) has at least two zeros in the interval \((0,2)\).

If \(f(x)\) be continuous in \([a, b]\) and the equation \(f(x)=0\) has only two roots \(\alpha\) and \(\beta(\alpha<\beta)\) in the interval \([a, b]\), then prove that \(f(x)\) retains the same sign in the interval \([\alpha, \beta]\).

Let \(f(x)\) be a continuous function defined for \(1 \leq x \leq 3\). If \(f(x)\) takes rational values for all \(x\) and \(f(2)=10\), then find \(f(1.5)\).

Let the function \(f(x)\) be continuous in the interval \([a, b]\) and \(a \leq f(x) \leq b \forall x \in[a, b]\). Prove that in this close interval there exists at least one point at which \(f(x)=x\). Explain this geometrically.

If \(f(x)\) is continuous and \(f(0)=f(1)\) then prove that there exists \(c \in\left[0, \frac{1}{2}\right]\) such that \(f(c)=f\left(c+\frac{1}{2}\right)\).

Let the function \(f(x)\) be continuous in the interval \([a, b]\). Prove that in this close interval there exists at least one point at which \(f(x)=\frac{f(a)+f(b)}{2}\).

Prove that if the function \(f(x)\) is continuous in the interval \((a, b)\) and \(x_{1}, x_{2}, \ldots \ldots \ldots, x_{n}\) are any values in this open interval, then we can always find a real number \(c\) in this open interval such that \(f(c)=\frac{f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots \ldots .+f\left(x_{n}\right)}{n}\)

If \(f(x)\) is continuous and \(f\left(\frac{9}{2}\right)=\frac{2}{9}\) then find \(\lim _{x \rightarrow 0} f\left(\frac{1-\cos 3 x}{x^{2}}\right)\)

If \(f(x)\) is continuous in \([0,1]\) and \(f\left(\frac{1}{3}\right)=1\) then find \(\lim _{n \rightarrow \infty} f\left(\frac{n}{\sqrt{9 n^{2}+1}}\right)\).

If \(f(x)\) and \(h(x)\) are continuous functions and \(g(x)=\lim _{n \rightarrow \infty} \frac{x^{m} f(x)+h(x)+1}{2 x^{m}+3 x+3}, x \neq 1\) and \(g(1)=\lim _{x \rightarrow 1}(\ln e x)^{\frac{2}{\ln x}}\) and \(g(x)\) is continuous at \(x=1\), then find the value of \(2 g(1)+2 f(1)-h(1)\).