Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as
$$ f(x)=2 x^3-3 x^2-12 x+1 $$
Which of the following statements is/are correct?
(Here, $\mathbb{R}$ is the set of real numbers.)
The function $y(t)$ satisfies
$$ t^2 y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0 $$
where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y^{\prime}(0)=1$ and $y^{\prime}(1)=-1$, the maximum value of $y(t)$ over $[0,1]$ is ___________ (rounded off to two decimal places).
Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval.
Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality
$$ \beta \leq \int_2^8 f(x) d x \leq \alpha $$
is guaranteed to hold.
Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?