Below are three graphs: the graph of a function $f$, the graph of its derivative, and the graph of its second derivative. We have deliberately omitted to show any scale for the graph of $f$, but the other two graphs are traceable.
#[Step 1: Intercepts][Paso 1: Intersecciones]#
#[The number of $x$-intercepts is][El número de intersecciones-$x$ es]# RADIO.
#[The number of $y$-intercepts is][El número de intersecciones-$y$ es]# RADIO.
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Step 2: Extrema:
Now enter a comma-separated list of the $x$-values of all the absolute and relative extrema in the following form:
($x$-coordinate, relmin/relmax/absmin/absmax)
with coordinates accurate to one decimal place. For instance, if there is a relative maximum at $x=-1.2$ and an asolute minimum at $x=2.2$, enter
(-1.2,relmax),(2.2,,absmin)
or dne if there are no extrema.
Hint To locate the extrema accurately, refer to the graph of the derivative.
#[Extrema][Extremos]#: BOX
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Step 3: Points of inflection:
Now enter a comma-separated list of the $x$-values of all points of inflection with coordinates accurate to one decimal place. Enter dne if there are no points of inflection.
Hint To locate the points of inflection accurately, refer to the graph of the second derivative.
#[Points of Inflection][Puntos de inflección]#: BOX
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Step 4: Singular points:
Now enter a comma-separated list of the $x$-values of all the singular points in the following form:
($x$-value, left limit,right limit)
with coordinates accurate to the nearest whole number. For instance, if there is a singular point at $x=-5$ with left limit $\infty$ and right limit $-\infty$ and another at $x=6$ with left and right limits both $\infty$, enter
(-5,inf,-inf),(6,inf,inf)
or dne if there are no singular points.
Hint To locate the $x$-values of the singular points, refer to the graph of the derivative.
#[Singular points][Puntos singulares]#: BOX
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#[Step 5: Behavior at infinity:][Paso 5: Comportamiento en el infinito:]#
#[Now enter the value(s) of the given limits, accurate to four decimal places. For $\infty$ use inf and for $-\infty$ use -inf][Ahora ingresa el (los) valor(es) de los límites dados con una precisión de cuatro decimales. Para $\infty$ usa inf y para $-\infty$ usa -inf]]#
$\displaystyle \lim_{x \to -\infty}f(x) =$ BOX
$\displaystyle \lim_{x \to -\infty}f(x) =$ BOX
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