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

Let $f(x)$ be a cubic polynomial with $f(1) = -10$,$f(-1) = 6$,and it has a local minima at $x = 1$. Also,$f'(x)$ has a local minima at $x = -1$. Then $f(3)$ is equal to:

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime}(0)$ is equal to

$A$ particle moving in a straight line starts from rest and the acceleration at any time $t$ is $a - kt^2$, where $a$ and $k$ are positive constants. The maximum velocity attained by the particle is

At what value of $x$ does the function $f(x) = x^x$ $(x > 0)$ attain its minimum value?

Let $a$ be a real number such that the function $f(x) = ax^2 + 6x - 15, x \in R$ is increasing in $(-\infty, \frac{3}{4})$ and decreasing in $(\frac{3}{4}, \infty)$. Then the function $g(x) = ax^2 - 6x + 15, x \in R$ has a: