Applications of Derivatives
Local Minima with Parameter Constraint
nta_pyq_2024_apr
Grade 12
Question:
Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $f(x)=1+x(\lambda^2-x^2)$ satisfies $\dfrac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha,\beta)$. Then $\alpha^2+\beta^2$ is equal to _________.
Step-by-Step Solution
Key Concept: Local min at $x=-\lambda/\sqrt{3}$. Condition $x\in(-3,-2)\Rightarrow2\sqrt{3}<\lambda<3\sqrt{3}$.
$\alpha^2+\beta^2=12+27=39$.
Correct Answer: 39