Question

The number of real roots of the polynomial \[ f(x) = x^{11} - 13x + 5 \] is ..................

Hint:

For odd-degree polynomials with positive leading coefficients, there is always at least one real root. Monotonicity of the derivative can confirm it is exactly one.

Answer 1

Correct Answer: 3

Step 1: Analyze the behavior of \(f(x)\).
As \(x \to \infty\), \(f(x) \to +\infty\); and as \(x \to -\infty\), \(f(x) \to -\infty\). Hence, the function must cross the x-axis at least once.
Step 2: Examine the derivative.
\[ f'(x) = 11x^{10} - 13. \] Set \(f'(x) = 0\) gives: \[ x^{10} = \frac{13}{11}. \] \[ x = \pm \left(\frac{13}{11}\right)^{1/10}. \] Thus, \(f'(x)\) changes sign once from negative to positive, confirming only one turning point.
Step 3: Sign of function values.
\(f(-\infty)<0\), \(f(\infty)>0\), and since \(f(x)\) changes sign only once, it crosses the x-axis only once. Final Answer: \[ \boxed{1} \]