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Found: 3

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Polynomials Quadratic polynomials

12–14 2.0

Find all functions $f (x)$ such that $f (2 x + 1) = 4 x^{2} + 14 x + 7$ .

Problem #PRU-111763

Problems Calculus Derivative Calculating a derivative Algebra Absolute value Inequalities containing absolute values Polynomials Quadratic polynomials

15–16 4.0

The quadratic trinomials $f (x)$ and $g (x)$ are such that $f^{'} (x) g^{'} (x) \geq | f (x) | + | g (x) |$ for all real $x$ . Prove that the product $f (x) g (x)$ is equal to the square of some trinomial.

Problem #PRU-78043

Problems Methods Methods of calculus Monotonicity and boundedness Pigeonhole principle Pigeonhole principle (other) Algebra Polynomials Quadratic polynomials

13–15 4.0

The expression $a x^{2} + b x + c$ is an exact fourth power for all integers $x$ . Prove that $a = b = 0$ .