The function $f(x) = x^3 + ax^2 + bx + c$ where $a^2 \leq 3b$ has:

Let $f(x) = 1 + 2x^2 + 2^2x^4 + \dots + 2^{10}x^{20}$. Then $f(x)$ has:

An open tank with a square bottom is to contain $4000 \ cm^3$ of liquid. Find the dimensions of the tank such that the surface area of the tank is minimum.

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$ for all $x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln(g(x))$ for all $x \in R$,then the number of points in $R$ at which $g$ has a local maximum is

Let $f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$. The number of points of local maxima of $f$ in the interval $(0, 2 \pi)$ is:

The set of value$(s)$ of $a$ for which the function $f(x) = \frac{a x^3}{3} + (a + 2) x^2 + (a - 1) x + 2$ possesses a negative point of inflection.