The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ $(a>0)$ attains its maximum and minimum at $p$ and $q$ respectively and $p^2=q$. Then,$a=$

Find the maximum profit that a company can make,if the profit function is given by $p(x) = 41 - 72x - 18x^{2}$. (in $units$)

$A$ wire of length $20$ units is divided into two parts such that the product of one part and the cube of the other part is maximum. Then,the product of these parts is:

Let the function $f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then,the set of all values of $p$ is $......$

Let $\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac{\pi}{6}\right)$. Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by $g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}$. Then,which of the following statements is/are $TRUE$?
$(A)$ The minimum value of $g(x)$ is $2^{7/6}$
$(B)$ The maximum value of $g(x)$ is $1 + 2^{1/3}$
$(C)$ The function $g(x)$ attains its maximum at more than one point
$(D)$ The function $g(x)$ attains its minimum at more than one point

If the function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$,where $a > 0$,attains its local maximum and local minimum values at $p$ and $q$ respectively,such that $p^2=q$,then $f(3)$ is equal to: