Regarding the function \(f(x)=x^{2}-5 x+1\) and \(g(x)=1-x-x^{3}\) on the set of real numbers, which of the following statements is NOT true?
Question and options
Regarding the function and on the set of real numbers, which of the following statements is NOT true?
- The graph of and intersect at three different points.
- The graph of and intersect at just one point with positive abscissa.
- The graph of and intersect at just one point with negative abscissa.
- The graph of and intersect at just one point with a positive -coordinate.
Answer and explanation
Correct answer: D - The graph of \(f\) and \(g\) intersect at just one point with a positive \(y\)-coordinate.
f = g gives x^3 + x^2 - 4x = 0, x = 0, (-1 +/- sqrt17)/2 ~ 1.56, -2.56: three points, one with positive and one with negative abscissa. y-values: f(0)=1>0, f(-2.56)~20.4>0, f(1.56)~-4.4<0, so two intersection points have a positive y-coordinate; the claim that just one does is false.
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