Applications of Derivatives
Application of Derivatives
nta_pyq_2025_apr
Grade 12
Question:
If the set of all values of $a$, for which the equation $5x^3 - 15x - a = 0$ has three distinct real roots, is the interval $(\alpha, \beta)$, then $\beta - 2\alpha$ is equal to ____.
Step-by-Step Solution
Key Concept: Let $h(x) = 5x^3 - 15x$. For $h(x) = a$ to have three distinct real roots, $a$ must lie strictly between the local minimum value and the local maximum value of $h$.
Let $h(x) = 5x^3 - 15x$. Then $h'(x) = 15x^2 - 15 = 15(x-1)(x+1)$.
Local maximum at $x = -1$: $h(-1) = -5+15 = 10$.
Local minimum at $x = 1$: $h(1) = 5-15 = -10$.
For three distinct roots, $a \in (-10, 10)$, so $\alpha = -10$, $\beta = 10$.
$$\beta - 2\alpha = 10 - 2(-10) = 10 + 20 = 30.$$
Correct Answer: 30