Prove that the root a of the polynomial \(P (x)\) has multiplicity greater than 1 if and only if \(P (a) = 0\) and \(P '(a) = 0\).
For which \(A\) and \(B\) does the polynomial \(Ax^{n + 1} + Bx^n + 1\) have the number \(x = 1\) at least two times as its root?
Find the largest and smallest values of the functions
a) \(f_1 (x) = a \cos x + b \sin x\); b) \(f_2 (x) = a \cos^2x + b \cos x \sin x + c \sin^2x\).
The Newton method (see Problem 61328) does not always allow us to approach the root of the equation \(f(x) = 0\). Find the initial condition \(x_0\) for the polynomial \(f(x) = x (x - 1)(x + 1)\) such that \(f(x_0) \neq x_0\) and \(x_2 = x_0\).
Prove the inequality: \[\frac{(b_1 + \dots b_n)^{b_1 + \dots b_n}}{(a_1 + \dots a_n)^{b_1 + \dots + b_n}}\leq \left(\frac{b_1}{a_1}\right)^{b_1}\dots \left( \frac{b_n}{a_n}\right)^{b_n}\] where all variables are considered positive.
Inequality of Jensen. Prove that if the function \(f (x)\) is convex upward on \([a, b]\), then for any distinct points \(x_1, x_2, \dots , x_n\) (\(n \geq 2\)) from \([a; b]\) and any positive \(\alpha_{1}, \alpha_{2}, \dots , \alpha_{n}\) such that \(\alpha_ {1} + \alpha_{2} + \dots + \alpha_{n} = 1\), the following inequality holds: \(f (\alpha_{1} x_1 + \dots + \alpha_{n} x_n) > \alpha_{1} f (x_1) + \dots + \alpha_{n} f (x_n)\).
Prove that the polynomial \(P (x)\) is divisible by its derivative if and only if \(P (x)\) has the form \(P(x) = a_n(x - x_0)^n\).
A polynomial of degree \(n > 1\) has \(n\) distinct roots \(x_1, x_2, \dots , x_n\). Its derivative has the roots \(y_1, y_2, \dots , y_{n-1}\). Prove the inequality \[\frac{x_1^2 + \dots + x_n^2}{n}> \frac{y_1^2 + \dots + y_n^2}{n}.\]
Solve the equation \(2 \sin \pi x / 2 - 2 \cos \pi x = x^5 + 10x - 54\).
Solve the equation \(f (f (x)) = f (x)\) if \(f(x) = \sqrt[5]{3 - x^3 - x}\).