Chapter 8

For what values of \(k\), one root of the function \((k-5) x^{2}-2 k x+k-4\) is smaller than 1 and the other root exceeds 2?

For what values of \(a\) does the function \(x^{2}+2(a-1) x+a+5\) has at least one positive root?

Let \(a, b, c\) be real. If \(a x^{2}+b x+c=0\) has two real roots \(\alpha\) and \(\beta\), where \(\alpha<-1\) and \(\beta>1\), then show that \(1+\frac{c}{a}+\left|\frac{b}{a}\right|<0\)

Prove that the value of \(a\) for which \(2 x^{2}-2(2 a+1) x+a(a+1)=0\) may have one root less than \(a\) and the other root greater than \(a\), are given by \(a>0\) or \(a<-1\).

Find all values of the parameter \(a\) for which the roots of the function \(x^{2}+x+a\) are real and exceed \(a\) ?

\(a, b, c\) are real numbers, \(a \neq 0\). If \(\alpha\) is a root of \(a^{2} x^{2}+b x+c=0, \beta\) is a root of \(a^{2} x^{2}-b x-c=0\) and \(0<\alpha<\beta\), then show that the equation \(a^{2} x^{2}+2 b x+2 c=0\) has a root \(\gamma\) that always lies between \(\alpha\) and \(\beta\).

For what values of \(k\), the function \(f(x)=k x^{3}-9 x^{2}+9 x+3\) is monotonically increasing in every interval?

For what values of \(k\), the function \(f(x)=x^{3}-9 k x^{2}+27 x+30\) is increasing on \(R\) ?

Find the minimum value of \((x-a)(x-b)\).

If the function \(f(x)=x^{2}-k x+5\) is increasing on \([2,4]\), then find the value of \(k\).

Find all numbers \(a\) for which the least value of the quadratic function \(4 x^{2}-4 a x+a^{2}-2 a+2\) in the interval \([0,2]\) is equal to 3? \\{

Find all real values of \(m\) for which the inequality \(m x^{2}+4 x+3 m+1>0\) is satisfied for all positive \(x ?\)

For what values of \(a\) does the inequality \(4^{x}-a \cdot 2^{x}-a+3 \leq 0\) has at least one solution?

For what values of \(a\) does the equation \(2 \log _{3}^{2} x-\left|\log _{3} x\right|+a=0\) possess i. four solutions. ii. three solutions. iii. two solutions. iv. one solution. \(\mathrm{v}_{4}\) no solution.

Solve the equation \(x^{2}-|x|+a=0\) for every real number \(a\).

For what values of \(a\) does the equation \(a \cdot 2^{x}+2^{-x}=5\) possess i. two solutions. ii. one solution. iii. no solution.

Find the range of the function \(f(x)=\frac{x^{2}-x+1}{x^{2}+x+1}\).

Find the range of the function \(f(x)=\frac{x+1}{x^{2}+x+1}\).

Find the range of the function \(\frac{x^{2}-2 x+4}{x^{2}+2 x+4}\).

For real values of \(x\), prove that the value of the expression \(\frac{11 x^{2}+12 x+6}{x^{2}+4 x+2}\) cannot lie between \(-5\) and 3 .

If \(x\) be real, prove that the expression \(\frac{x+2}{2 x^{2}+3 x+6}\) takes all values in the interval \(\left[-\frac{1}{13}, \frac{1}{3}\right]\).

Prove that for real values of \(x\) the expression \(\frac{(x-1)(x+3)}{(x-2)(x+4)}\) cannot lie between \(\frac{4}{9}\) and \(1 .\)

If \(x\) is real, show that the expression \(\frac{x^{2}+2 x-11}{x-3}\) takes all values which do not lie between 4 and 12 .

If \(x\) is real, find the maximum and minimum values of \(\frac{x^{2}+14 x+9}{x^{2}+2 x+3}\).

Show that \(\frac{x^{2}-3 x+4}{x^{2}+3 x+4}\) can never be greater than 7 nor less than \(\frac{1}{7}\) for real values of \(x\).

Show that the expression \(\frac{p x^{2}+3 x-4}{p+3 x-4 x^{2}}\) will be capable of all real values when \(x\) is real, provided that \(p\) has any value between 1 and 7 .

For what values of \(a\) is the inequality \(\frac{x^{2}+a x-1}{2 x^{2}-2 x+3}<1\) fulfilled for all \(x\) ?

For what values of \(a\) is the inequality \(-6<\frac{2 x^{2}+a x-4}{x^{2}-x+1}<4\) fulfilled for all \(x\) ?

Find the value of \(\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}\)

If \(f(x)\) is a polynomial function satisfying \(f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)\) and \(f(3)=28\), then find \(f(4)\).

A function \(f: R \rightarrow R\) is defined by \(f(x)=\frac{\alpha x^{2}+6 x-8}{\alpha+6 x-8 x^{2}}\), find the interval of values of \(\alpha\) for which \(f\) is onto. Is the function one-to-one for \(\alpha=3\) ? Justify your answer.

Let \(c\) be a fixed real number. Show that a root of the polynomial \(P(x)=x(x+1)(x+2) \cdots \cdots(x+2009)-c\) can have multiplicity at most 2 .