The function $f(x) = x^{-x}, (x \in R)$ attains a maximum value at $x =$

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:

$A$ population $p(t)$ of $1000$ bacteria introduced into a nutrient medium grows according to the relation $p(t) = 1000 + \frac{1000t}{100 + t^2}$. The maximum size of this bacterial population is

The shortest distance between the point $\left( \frac{3}{2}, 0 \right)$ and the curve $y = \sqrt{x}, (x > 0)$ is:

Let $A$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is ..................

$A$ ball is thrown in the air. Its height at any time $t$ is given by $h = 3 + 14t - 5t^2$. What is the maximum height it can reach?