MathLabs

Problem 3

Determine the minimum of a2+b2a^2+b^2 over real a,ba,b for which x4+ax3+bx2+ax+1=0x^4+ax^3+bx^2+ax+1=0 has at least one real solution.
Step 5 of 5: Verify equality
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In plain words

The factorization confirms that the boundary point is attainable.

x4+45x3−25x2+45x+1=(x+1)2(x2−65x+1)x^4+\frac45x^3-\frac25x^2+\frac45x+1=(x+1)^2(x^2-\frac65x+1)
Detailed analysis

The factorization has the real root x=−1x=-1, so the minimum is indeed 4/5\boxed{4/5}.