Question

Let a and b be real constants such that \[x^4 ax^3 3x^2 bx 1 \ge 0\]for all real numbers x. Find the largest possible value of a^2 b^2.

Answer 1

Answer:

Infinity

Step-by-step explanation:

Since $x^4 ax^3 3x^2 bx 1 \ge 0$ for all real numbers x, this property is also true for x=1, which tells us that

$1^4 a1^3 3\cdot1^2 b\cdot 1=3ab\ge0$

On the other hand, note that for all real numbers x, it holds that

$x^4x^3x^2x\ge 0$

Therefore, if

$3ab\geq0$

we have tat

$3ab x^{4}x^{3}x^{2}x^{1}1=x^4ax^3 3x^2bx1\ge0$

The last reasoning tells us that the property $x^4 ax^3 3x^2 bx 1 \ge 0$ holds for all real numbers x if an only if $ab\ge0$

Therefore, we can choose arbitrary constants a and b as long as

$ab\ge0$

We can choose a and b such that both positive, both negative or one of the two constants is equal two zero. In the first two cases

$a^2b^2$  

can get as big as we want, depending on the constants, and we are done.