Find the maximum and minimum values for the function given by $g(x)=-|x+1|+3$.

An open metallic tank is to be constructed with a square base and vertical sides,having a volume of $500 \,m^3$. The dimensions of the tank for the minimum area of the sheet metal used in its construction are:

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\mathop {\lim }\limits_{x \to 0} \left[ {1 + \frac{{f(x)}}{{{x^2}}}} \right] = 3$,then $f(2)$ is equal to:

If $f(x) = 2x^6 - 3$,then:

Let the function $f: (0, \pi) \rightarrow R$ be defined by $f(\theta) = (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^4$. Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_1 \pi, \dots, \lambda_r \pi\}$,where $0 < \lambda_1 < \dots < \lambda_r < 1$. Then the value of $\lambda_1 + \dots + \lambda_r$ is:

The curve $y = 2x^3 + ax^2 + bx + c$ passes through the origin,and the tangents at $x = -1$ and $x = 2$ are parallel to the $X$-axis. Then the values of $a, b,$ and $c$ are respectively: