Correct option is B
Given:
The polynomial is f(x) = x³ + 3x - 2023.
We need to determine how many real roots this polynomial has.
Solution:
Behavior of the Polynomial:
- The polynomial f(x) = x³ + 3x - 2023 is a cubic function, so it is continuous and differentiable everywhere.
- Cubic functions always have at least one real root because they are continuous and the degree is odd.
Derivative Analysis:
- The derivative is f'(x) = 3x² + 3, which is always positive (f'(x) > 0 for all x).
- Since the derivative is positive, f(x) is strictly increasing.
Number of Real Roots:
- A strictly increasing function can cross the x-axis at most once.
- Therefore, f(x) = x³ + 3x - 2023 has exactly one real root.
Final Answer: (b) 1
Another method:
The function is given as:
f(x) = x³ + 3x - 2023.
The derivative of the function is:
f'(x) = 3(x² + 1), which is greater than 0 for all x belonging to R (real numbers).
Since f'(x) > 0 for all x in R, the function f(x) is strictly increasing on R.
This implies that f(x) maps the interval (-∞, ∞) to the interval (f(-∞), f(∞)) as a one-to-one and onto function.
Therefore, f(x): R → R is a bijective map, meaning it has exactly one unique solution where f(x) = 0.
- As x → -∞, f(x) → -∞.
- As x → ∞, f(x) → ∞.
Conclusion: The polynomial f(x) = 0 has a unique real root.
