The question is about maximum possible value. We need to maximize a expression which is related to the roots of the polynomial. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 22: x^{4} โ 4x^{3} + ax^{2} โ bx = 1 = 0 has positive real roots. What is the maximum possible value of a + b?

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10

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Fabulous question.

If we have an equation of the form ax^{4} + bx^{3} + cx^{2} + dx + e = 0 with Roots p, q, r and s.

We can say sum of the roots p + q + r + s = \\frac{-b}{a}\\)

Sum of the products taken two at a time pq + pr + ps + qr + qs + rs = ๐\\frac{c}{a}\\) .

Sum of the products taken three at a time pqr + pqs + prs + qrs = \\frac{-d}{a}\\)

Product of the roots pqrs = ๐\\frac{e}{a}\\) .

Note 1: We alternate between - and +

Note 2: Start with the immediate lower power for sum of roots, stick a negative symbol and then alternate.

So, if p, q, r, s were roots of this equation

p + q + r + s = 4,

pqrs = 1

Or, Arithmetic mean of p,q, r, s = 1 and geometric mean of pqrs = 1.

P, q, r s are positive real numbers. AM = GM. What does this mean?

This means that all 4 numbers are equal.

Or, this expression is (x-1)^{4}

a = pq + pr + ps + qr + qs + rs = 6

-(-b) = pqr + pqs + prs + qrs = 4

a = 6, b = 4. a + b = 10

The question is **"What is the maximum possible value of a + b?"**

Choice D is the correct answer.

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