MHB Maximizing Real Roots: Find Biggest Possible Value of a | POTW #199

[anemone]

Here is this week's POTW:

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Let $a,\,b,\,c,\,d$ be real numbers such that the equation $x^5-20x^4+ax^3+bx^2+cx+d=0$ has real roots only. Find the biggest possible value of $a$.

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[anemone]

Congratulations to kaliprasad for his correct solution.:)

You can find the proposed solution below:

Spoiler

Let $f(x)=x^5-20x^4+ax^3+bx^2+cx+d$. If $f$ has all real roots, then the function of the third derivative of $f$ must have two real roots.

$f'(x)=5x^4-80x^3+3ax^2+2bx+c$

$f''(x)=20x^3-240x^2+6ax+2b$

$f'''(x)=60x^2-480x+6a$

If $f'''(x)=60x^2-480x+6a$ has two real roots, then its discriminant must be greater than or equal to zero:

$(-480)^2-4(60)(6a)\ge 0$

$a\le 160$

Therefore the biggest possible value of $a=160$.