Derive from the theorem in question 61013 that \(\sqrt{17}\) is an irrational number.
For a given polynomial \(P (x)\) we describe a method that allows us to construct a polynomial \(R (x)\) that has the same roots as \(P (x)\), but all multiplicities of 1. Set \(Q (x) = (P(x), P'(x))\) and \(R (x) = P (x) Q^{-1} (x)\). Prove that
a) all the roots of the polynomial \(P (x)\) are the roots of \(R (x)\);
b) the polynomial \(R (x)\) has no multiple roots.
Construct the polynomial \(R (x)\) from the problem 61019 if:
a) \(P (x) = x^6 - 6x^4 - 4x^3 + 9x^2 + 12x + 4\);
b)\(P (x) = x^5 + x^4 - 2x^3 - 2x^2 + x + 1\).
Prove that for \(n> 0\) the polynomial \(nx^{n + 1} - (n + 1) x^n + 1\) is divisible by \((x - 1)^2\).
An iterative polyline serves as a geometric interpretation of the iteration process. To construct it, on the \(Oxy\) plane, the graph of the function \(f (x)\) is drawn and the bisector of the coordinate angle is drawn, as is the straight line \(y = x\). Then on the graph of the function the points \[A_0 (x_0, f (x_0)), A_1 (x_1, f (x_1)), \dots, A_n (x_n, f (x_n)), \dots\] are noted and on the bisector of the coordinate angle – the points \[B_0 (x_0, x_0), B_1 (x_1, x_1), \dots , B_n (x_n, x_n), \dots.\] The polygonal line \(B_0A_0B_1A_1 \dots B_nA_n \dots\) is called iterative.
Construct an iterative polyline from the following information:
a) \(f (x) = 1 + x/2\), \(x_0 = 0\), \(x_0 = 8\);
b) \(f (x) = 1/x\), \(x_0 = 2\);
c) \(f (x) = 2x - 1\), \(x_0 = 0\), \(x_0 = 1{,}125\);
d) \(f (x) = - 3x/2 + 6\), \(x_0 = 5/2\);
e) \(f (x) = x^2 + 3x - 3\), \(x_0 = 1\), \(x_0 = 0{,}99\), \(x_0 = 1{,}01\);
f) \(f (x) = \sqrt{1 + x}\), \(x_0 = 0\), \(x_0 = 8\);
g) \(f (x) = x^3/3 - 5x^2/x + 25x/6 + 3\), \(x_0 = 3\).
The sequence of numbers \(a_n\) is given by the conditions \(a_1 = 1\), \(a_{n + 1} = a_n + 1/a^2_n\) (\(n \geq 1\)).
Is it true that this sequence is limited?
The numbers \(a_1, a_2, \dots , a_k\) are such that the equality \(\lim\limits_{n\to\infty} (x_n + a_1x_{n - 1} + \dots + a_kx_{n - k}) = 0\) is possible only for those sequences \(\{x_n\}\) for which \(\lim\limits_{n\to\infty} x_n = 0\). Prove that all the roots of the polynomial P \((\lambda) = \lambda^k + a_1 \lambda^{k-1} + a_2 \lambda^{k -2} + \dots + a_k\) are modulo less than 1.
Prove that for a monotonically increasing function \(f (x)\) the equations \(x = f (f (x))\) and \(x = f (x)\) are equivalent.
Prove that the tangent to the graph of the function \(f (x)\), constructed at coordinates \((x_0, f (x_0))\) intersects the \(Ox\) axis at the coordinate: \(x_0 -\frac{f(x_0)}{f'(x_0)}\).
Prove that if the function \(f (x)\) is convex upwards on the line \([a, b]\), then for any distinct points \(x_1, x_2\) in \([a; b]\) and for any positive \(\alpha_{1}, \alpha_{2}\) such that \(\alpha_{1} + \alpha_ {2} = 1\) the following inequality holds: \(f(\alpha_1 x_1 + \alpha_2 x_2 ) > \alpha_1 f (x_1) + \alpha_2 f(x_2)\).