The maximum value of $x^4 y^4$ given the constraint $a^2 x^4 + b^2 y^4 = c^6$ is

The least value of $\alpha \in R$ for which $4 \alpha x^2 + \frac{1}{x} \geq 1$,for all $x > 0$,is

If a line is moving between the coordinate axes such that the sum of the intercepts made by it on the coordinate axes is always $12$,then the equation of that line which forms a triangle of maximum area with the coordinate axes is

The number of distinct real roots of $x^{4}-4x+1=0$ is

Find the maximum value of $2x^{3}-24x+107$ in the interval $[1,3]$. Also,find the maximum value of the same function in the interval $[-3,-1]$.

Find the absolute maximum and minimum values of the function $f$ given by $f(x) = 12x^{\frac{4}{3}} - 6x^{\frac{1}{3}}$ for $x \in [-1, 1]$.