$$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable
function such that $$\left| {f\left( x \right)} \right| \le 1$$ and $$f'(x)=g(x).$$
If $${f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.$$ Prove that there exists some $$c \in \left( { - 3,3} \right)$$
such that $$g(c).g''(c)<0.$$
Answer
Solve it.
2
IIT-JEE 2005
Subjective
If $$\left[ {\matrix{
{4{a^2}} & {4a} & 1 \cr
{4{b^2}} & {4b} & 1 \cr
{4{c^2}} & {4c} & 1 \cr
} } \right]\left[ {\matrix{
{f\left( { - 1} \right)} \cr
{f\left( 1 \right)} \cr
{f\left( 2 \right)} \cr
} } \right] = \left[ {\matrix{
{3{a^2} + 3a} \cr
{3{b^2} + 3b} \cr
{3{c^2} + 3c} \cr
} } \right],\,\,f\left( x \right)$$ is a quadratic
function and its maximum value occurs at a point $$V$$. $$A$$ is a point of intersection of $$y=f(x)$$ with $$x$$-axis and point $$B$$ is such that chord $$AB$$ subtends a right angle at $$V$$. Find the area enclosed by $$f(x)$$ and chord $$AB$$.
Answer
$${{125} \over 3}$$ sq. units
3
IIT-JEE 2005
Subjective
Find the area bounded by the curves $${x^2} = y,{x^2} = - y$$ and $${y^2} = 4x - 3.$$