The shortest distance of the point $(0,0)$ from the curve $y = e^x + e^{-x}$ is-

The denominator of a fraction is $16$ more than the square of the numerator. Find the least value of the fraction.

The area bounded by the tangents of the curve given by $y = \sin \theta \cos^2 \theta$ and $x = \sin^2 \theta \cos \theta$,which are parallel to the coordinate axes (excluding the axes themselves),is:

The abscissae of the points of the curve $y = x(x - 2)(x - 4)$ where the tangents are parallel to the $x$-axis are obtained as:

Let the sum of the maximum and the minimum values of the function $f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8}$ be $\frac{m}{n}$,where $\gcd(m, n) = 1$. Then $m + n$ is equal to :

The maximum value of $f(x) = \frac{x}{4 + x + x^2}$ on the interval $[-1, 1]$ is