Functions
Cubic with 3 positive roots — properties of derived function
MJAT_TS6_P1
Grade 12
Question:
The equation $x^3\sin q-(\sin q+2)x^2+6x-4=0$, $q\in(-\pi/2,\pi/2)$ has 3 positive roots. Let $f(\sin q)=\dfrac{9\sin^2 q-4\sin q+3}{(1-\cos q)(2\cos q-6\sin q-3\sin 2q+2)}$. Then which is/are true?
A) $f(t)$ attains its maximum in its domain
B) $f(t)$ attains its minimum in its domain
C) $f(t)$ attains at least one local minimum in its domain
D) $f(t)$ attains at least one local maximum in its domain
Step-by-Step Solution
Key Concept: For 3 positive roots of the cubic: by Vieta and AM-GM analysis, determine the constraint on $\sin q$. Then find the domain of $f$ and analyse its extrema.
A ✓, C ✓. Answer: A, C.
Correct Answer: AC