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If [math]\alpha[/math] and [math]\beta[/math] are roots of equation [math]x^2-2px+2p^4[/math], then what is the maximum value of [math]\alpha^2+\beta^2[/math]?
How do we find the maximum value of [math]\alpha^2+\beta^2[/math]?
We can differentiate [math]4p^2-4p^4[/math] with respect to [math]p[/math] and equate it to zero. 
On solving, we get [math]p=\frac{1}{\sqrt{2}}[/math]. Substituting this value, we get the maximum value of [math]\alpha^2+\beta^2[/math] to be [math]2[/math].