Question

The equation \( x^{3} + x - 1 = 0 \) has

• A. no real root.
• B. exactly two real roots.
• C. exactly one real root.
• D. more than two real roots.

Hint:

Calculus Tip:If a continuous function's derivative is strictly positive ($f'(x)>0$) or strictly negative ($f'(x)<0$) on an interval, it implies the function is one-to-one (injective) and can cross any horizontal line at most once.The Intermediate Value Theorem is the standard tool for proving the existence of a root in a specific interval.

Answer 1

The Correct Option is C

Concept: Calculus - Application of Derivatives (Monotonicity) and Intermediate Value Theorem.

Step 1: Define the function to analyze. Let $f(x) = x^3 + x - 1$. Finding the roots of the equation means finding the values of $x$ for which $f(x) = 0$.

Step 2: Find the first derivative of the function. Differentiate $f(x)$ with respect to $x$ to determine the behavior of the function. $f'(x) = \frac{d}{dx}(x^3 + x - 1) = 3x^2 + 1$.

Step 3: Analyze the sign of the derivative. For any real number $x$, the square of that number is non-negative ($x^2 \ge 0$). Multiplying by 3 maintains this non-negativity ($3x^2 \ge 0$).
Adding 1 guarantees that the expression is strictly positive: $3x^2 + 1 \ge 1>0$. Thus, $f'(x)>0$ for all $x \in \mathbb{R}$.

Step 4: Determine the monotonicity of the function. Because $f'(x)>0$ for all real numbers, the function $f(x)$ is strictly increasing over its entire domain.
A strictly increasing continuous function can cross the x-axis at most one time. Therefore, it has at most one real root.

Step 5: Confirm the existence of a root using the Intermediate Value Theorem. Let's test some simple values. $f(0) = 0^3 + 0 - 1 = -1$ (which is negative). $f(1) = 1^3 + 1 - 1 = 1$ (which is positive). Because $f(x)$ is a polynomial (and therefore continuous), and it changes sign between $x=0$ and $x=1$, the Intermediate Value Theorem guarantees there is at least one root in the interval $(0, 1)$. Combined with Step 4, there is exactly one real root. $$ \therefore \text{The given equation has exactly one real root.} $$