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Chapter 2: Problem 2

Determine how the number of real roots of the equation $$ g(x)=4 x^{3}-17 x^{2}+10 x+k=0 $$ depends upon \(k\). Are there any cases for which the equation has exactly two distinct real roots?

Short Answer

Expert verified
The cubic polynomial does not ever have exactly two distinct real roots, it can either have one real root or three real roots depending on the value of \(k\).

Step by step solution

01

Find the derivative of the function

The first step in finding real roots is to find the derivative of the given function. Thus the derivative, \(g'(x)\), of \(g(x)=4x^3 - 17x^2 + 10x + k\) is \(g'(x) = 12x^2 - 34x + 10\). This derivative can be set equal to zero to find potential extrema.
02

Find the roots of the derivative function

Next, we set the derivative equal to zero to find the stationary points, \(12x^2 - 34x + 10 = 0\). Solve the quadratic equation to find two roots, where \(x_1\) and \(x_2\) are roots of the equation. They split the number line into three parts.
03

Test the intervals between the roots

Take test points from each of these intervals and substitute them in the derivative \(g'(x)\). If \(g'(x)\) is positive, then in this interval the function \(g(x)\) is increasing. If \(g'(x)\) is negative, then in this interval the function \(g(x)\) is decreasing.
04

Find the relation between roots of \(g(x)\) and roots of \(g'(x)\)

Based on the results from Step 3, if \(g(x)\) has a root in an interval where \(g(x)\) is decreasing or increasing, then \(g(x)\) has exactly one root in this interval. Therefore, \(g(x)\) will have three real roots if \(k\) is less than the minimum of the function or greater than the maximum. If \(k\) equals to the extremum values, it will have two real roots. Otherwise, \(g(x)\) will have just one real root.
05

Analyze if two distinct real roots is possible

Considering that this is a cubic function, we know that it can't have exactly two distinct real roots. This is because if we consider a horizontal line \(y=c\), if it intersects the cubic at two points it must intersect at a third since between any two roots there must be a local minimum or maximum. Therefore, a cubic function can't have exactly two distinct real roots.

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If a sequence of terms, \(u_{n}\), satisfies the recurrence relation \(u_{n+1}=(1-x) u_{n}+\) \(n x\), with \(u_{1}=0\), show, by induction, that, for \(n \geq 1\), $$ u_{n}=\frac{1}{x}\left[n x-1+(1-x)^{n}\right] $$

Prove by induction that $$ \sum_{r=1}^{n} r=\frac{1}{2} n(n+1) \quad \text { and } \quad \sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2} $$

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