Applications of Derivatives
Local Extrema — Vieta's Relations
nta_pyq_2024_apr
Grade 12
Question:
If the function $f(x)=2x^3-9x^2+12a^2x+1$, $a>0$, has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation:
$x^2-6x+8=0$
$x^2+6x+8=0$
$8x^2+6x-1=0$
$8x^2-6x+1=0$
Step-by-Step Solution
Key Concept: $\alpha+\alpha^2=3a$, $\alpha^3=2a^2$. Solving: $a=2$. Quadratic: $x^2-6x+8=0$.
$a=2$. Roots satisfy $x^2-6x+8=0$.
Correct Answer: 1