If \( x, y \) are two positive integers such that \( x + y = 20 \) and the maximum value of \( x^3 y \) is \( k \) at \( x = a, y = \beta \), then \( \frac{k}{\alpha^2 \beta^2} = ? \)
If the function f(x) = \(\sqrt{x^2 - 4}\) satisfies the Lagrange’s Mean Value Theorem on \([2, 4]\), then the value of \( C \) is}
The angle between the curves \( y^2 = 2x \) and \( x^2 + y^2 = 8 \) is
If \( T = 2\pi \sqrt{\frac{L}{g}} \), \( g \) is a constant and the relative error in \( T \) is \( k \) times to the percentage error in \( L \), then \( \frac{1}{k} = \) ?
If \( y = x - x^2 \), then the rate of change of \( y^2 \) with respect to \( x^2 \) at \( x = 2 \) is:
For \( x<0 \), \( \frac{d}{dx} [|x|^x] \) is given by: