Question:
The real number $k$ for which the equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0, 1]$
The real number $k$ for which the equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0, 1]$
Updated On: Feb 14, 2025
- lies between $2$ and $3$
- lies between $-1$ and $0$
- does not exist
- lies between $1$ and $2$
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The Correct Option is C
Solution and Explanation
Let $f (x) = 2x^3 + 3x + k, f'(x) = 6x^2 + 3 > 0$
Thus $f$ is strictly increasing. Hence it has atmost one real root. But a polynomial equation of odd degree has atleast one root.
Thus the equation has exactly one root.
Then the two distinct' roots; in any interval whatsoever is an impossibility. No such does not
exists.
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Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root