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?

Abscissa = $x$-coordinate

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Intersections satisfy $f(x)=g(x)$:

$x^2-5x+1 = 1-x-x^3$

$x^3+x^2-4x = 0$

$x(x^2+x-4)=0$

So $x=0$ or $x=\dfrac{-1\pm\sqrt{17}}{2}$, giving

$x_1 \approx -2.56,\qquad x_2 = 0,\qquad x_3 \approx 1.56$

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Checking each statement

Three different points: three distinct real roots, so true.

One point with positive abscissa: only $x_3\approx 1.56$ is positive, so true.

One point with negative abscissa: only $x_1\approx -2.56$ is negative, so true.

One point above the $x$-axis: need the $y$-values. For the two irrational roots, use $x^2 = 4-x$, so

$y = x^2-5x+1 = (4-x)-5x+1 = 5-6x$

• $x_1\approx -2.56 \Rightarrow y \approx 20.4$ (above)
• $x_2 = 0 \Rightarrow y = 1$ (above)
• $x_3\approx 1.56 \Rightarrow y \approx -4.37$ (below)

Two of the three intersection points lie above the $x$-axis, not one.