If the curve $y = x^4 + 4rx + 3s$ , where r and s are real numbers; lies entirely above the x - axis, then: (a) $r^2 s^3$ (c) $r^4 s^3$

Question

If the curve $y = x^4 + 4rx + 3s$ , where r and s are real numbers; lies entirely above the x - axis, then: (a) $r^2 < s^3$ (b) $r^2 > s^3$ (c) $r^4 s^3$

vishweshshrimali5 · Answer

@ganeshie8 @Miracrown @mukushla

vishweshshrimali5 · Answer

What I did was this:

For the curve to lie entirely above the x-axis; its minimum value must be positive.

So, using maxima - minima principle,

$y' = 4x^3 + 4r$

Putting y' = 0

$4x^3 + 4r = 0$

$\large{\implies x^3 = -r}$

$\large{\implies x = -r^{\cfrac{1}{3}}}$

Now, 

$\large{y'' = 12x^2}$

which will always be positive irrespective of value of x.

Thus, there is minima at $\large{x = -r^{\cfrac{1}{3}}}$.

So,

$\large{y_{min} = r^{\cfrac{4}{3}} - 4r^{\cfrac{4}{3}} + 3s}$

vishweshshrimali5 · Answer

Now, $y_{min} > 0$

Thus,

$\large{y_{min} = r^{\cfrac{4}{3}} - 4r^{\cfrac{4}{3}} + 3s > 0}$

$\large{\implies-3r^{\cfrac{4}{3}} + 3s > 0}$

$\large{\implies s > r^{\cfrac{4}{3}}}$

$\large{\implies s^3 > r^4}$

Thus, option (c) should be correct.

Any alternative method from your side ?

vishweshshrimali5 · Answer

@mathslover

vishweshshrimali5 · Answer

@dan815

dan815 · Answer

to find s and r such that only complex roots exist

mathslover · Answer

That was such a great method.. @vishweshshrimali5 great application of maxima and minima. Awesome!

dan815 · Answer

yeah ^ :)