Range of an expression related to coefficients of a quadratic equationLet ax2+bx+c=0 be a quadratic equation such that both the roots lie in [0,1]. What is the range of the expression (a−b)(2a−b)a(a−b+c)?

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zalutaloj9a0f · Accepted Answer

(a−b)(2a−b)a(a−b+c)=(1+s)(2+s)1+s+pwhere s=−ba is the root sum and p=ca is the root product.s∈[0,2] and for a fixed s, p attains a maximum of s24 (when the roots are equal) and a minimum of 0 (if s≤1) or s−1 (if s≥1). Accordingly, to minimise the expression we must p=s24, whereupon it becomes4⋅(1+s)(2+s)(s+2)2=4⋅1+s2+sand clearly this is minimised at s=0, i.e. the lower bound on range is 2. To maximise the expression we consider two cases:- If s∈[0,1] we set p=0 and the expression simplifies to 2+s, whose maximum is 3.- If s∈[1,2] we set p=s−1 and the expression simplifies to (1+s)(2+s)2s=1s+32+s2, which is decreasing on [1,2] and increasing otherwise. Thus we check the value at s=2, which is also 3, and conclude that the maximum is still 3.Hence the expression range is [2, 3].

Range of an expression related to coefficients of

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