If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $$a>0,$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively such that $${p^2} = q$$ , then $$a$$ equals
Also $$f(x)$$ is continuous and differentiable in $$\left[ {0,1} \right]$$ and
$$\left[ {0,1\left[ {.\,\,} \right.} \right.$$ So by Rolle's theorem $$f'\left( x \right) = 0.$$
i.e $$\,\,a{x^2} + bx + c = 0$$ has at least one root in $$\left[ {0,1} \right].$$
3
AIEEE 2002
MCQ (Single Correct Answer)
The maximum distance from origin of a point on the curve
$$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$
$$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$$ both $$a,b > 0$$ is
A
$$a-b$$
B
$$a+b$$
C
$$\sqrt {{a^2} + {b^2}} $$
D
$$\sqrt {{a^2} - {b^2}} $$
Explanation
Distance of origin from $$\left( {x,y} \right) = \sqrt {{x^2} + {y^2}} $$