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For $f(x)=4x^3-6x^2-72x+30$, the function is decreasing on the interval
A$(-\infty,-2)$
B$(3,\infty)$
C$(-2,3)$
D$(-2,\infty)$
Answer & Solution
Correct answer: C. $(-2,3)$
1. $f'(x)=12x^2-12x-72=12(x-3)(x+2)$.
2. $f'(x)=0$ at $x=-2,3$.
3. On $(-2,3)$: $(x-3)<0$, $(x+2)>0$, so $f'(x)<0$ (decreasing).
4. On the outer intervals $f'(x)>0$ (increasing), ruling out A and B.
_Source: NCERT Class 12 Mathematics Ch 6 "Application of Derivatives", p.9_
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