Let f be a nonconstant polynomial of degree k and let g ∶ \(\mathbb{R}\) → \(\mathbb{R}\) be a bounded continuous function. Which of the following statements is necessarily true?
There exists x 0 ∈ \(\mathbb{R}\) s uch that f(x 0 ) = g(x 0 ) if k is odd
For odd k, the polynomial tends to opposite infinities at the two ends of the real line. Subtracting bounded continuous g preserves those opposite signs; the intermediate value theorem gives a zero of f−g. For even k an intersection need not exist: f(x) = x² and g(x) = −1 is a counterexample.
Important Questions from Real Number System
Let \(\mathbb{N}\) = {1, 2, 3, ...} be the set of natural numbers. Which of the following functions from \(\mathbb{N}\) × \(\mathbb{N}\) to \(\mathbb{N}\) are injective?
Let x, y be real numbers such that 0 < y ≤ x and let n be a positive integer. Which of the following statements are true?
Let x be a real number. Which of the following statements are true?
