Applications of Derivatives
Maxima and Minima
nta_pyq_2024_apr
Grade 12
Question:
Let $f(x) = 3\sqrt{x-2} + \sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2 + 2\beta^2$ is equal to:
Step-by-Step Solution
Key Concept: Domain: $x\in[2,4]$. Substitute $x=2\sin^2\theta+4\cos^2\theta$ so $f=3\sqrt{2}|\cos\theta|+\sqrt{2}|\sin\theta|$. By Cauchy-Schwarz, max $=\sqrt{(9+1)\cdot2}=\sqrt{20}$, min $=\sqrt{2}$.
$\alpha=\sqrt{2}$, $\beta=\sqrt{20}$. $\alpha^2+2\beta^2=2+2\times20=42$.
Correct Answer: 1