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 ..................

$x^x$ has a stationary point at

Find the local maximum and local minimum values for the function $h(x) = \sin x + \cos x$ in the interval $0 < x < \frac{\pi}{2}$.

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

The minimum value of the function $f(x) = 2x^2 - \ln|x|$ for $x \geq 1$ is:

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: