Classify the turning points of a rational curve

For $y=x/(x^2+1)$, the quotient rule gives $y^{\prime}=[(x^2+1)-2x^2]/(x^2+1)^2=(1-x^2)/(x^2+1)^2$. The denominator is positive for every real $x$. Thus $y^{\prime}=0$ exactly at $x=\pm1$, giving $(1,1/2)$ and $(-1,-1/2)$. The Teacher supplies the blue quotient-rule and coordinate working missing from the Student space.

At $x=1$, the derivative signs immediately left, at, and right are $+,0,-$: the tangent rises, is horizontal, then falls. Hence $(1,1/2)$ is a local maximum. At $x=-1$, the signs are $-,0,+$: falling, horizontal, rising, so $(-1,-1/2)$ is a local minimum. These retain both printed sign/tangent tables. The Teacher advises an analytical method when the question says “exact”, “by using differentiation” or “without the use of a G.C.”.

The printed quotient-rule second derivative is $[(x^2+1)^2(-2x)-(1-x^2)2(x^2+1)(2x)]/(x^2+1)^4$. It simplifies to $2x(x^2-3)/(x^2+1)^3$. At $x=1$, $y^{\prime\prime}=-1/2<0$, confirming the maximum; at $x=-1$, $y^{\prime\prime}=1/2>0$, confirming the minimum. The distinct Teacher note recommends the first derivative test when obtaining the second derivative is not easy.

The self-reading alternative begins Y=, enter $Y_1=X/(X^2+1)$, then GRAPH. Use WINDOW to magnify the region if needed. The source screen shows the curve crossing the origin, with a trough near $X=-1$ and a peak near $X=1$. This is numerical exploration; the analytical result above supplies the exact coordinates.

Press 2nd → TRACE, select 3:minimum, place the cursor slightly left of the minimum and press ENTER, then slightly right and ENTER, then ENTER again for the guess. The printed result is $(-1,-0.5)$; the calculator screen’s x-value is approximate, rather than exactly $-1$.

Press 2nd → TRACE, select 4:maximum and similarly supply left bound, right bound and guess around the maximum, pressing ENTER at each stage. The printed result is $(1,0.5)$. Both originals’ step 6 instructions accidentally repeat “minimum point”; the selected menu and result show that the intended target here is the maximum. That correction is stated separately from the unchanged native transcription. No operation of a physical calculator is claimed.