# `f(x) = ((x + 4)(x - 3)^2)/((x^4)(x - 1))` Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing...

`f(x) = ((x + 4)(x - 3)^2)/((x^4)(x - 1))` Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values.

gsarora17 | Certified Educator

`f(x)=((x+4)(x-3)^2)/(x^4(x-1))`

Vertical asymptotes are real zeros of the denominator of the function.

`x^4(x-1)=0`

`x=0 , x=1`

Vertical asymptotes are at x=0 and x=1

Degree of numerator =3``

Degree of denominator=5

Since degree of denominator is `>` degree of numerator,

so Horizontal asymptote is the x-axis. HA is y=0

See the attached image and links. f(x) is plotted in several ranges to have clarity.

From the graph f has no y intercept , x intercepts x=-4 , 3

Function has three maximum values,

f(`~~` -5.5)`~~` 0.02

f(`~~` 5) `~~` 0.015

f(`~~` 0.825) `~~` -275

Function has one Minimum value

f(3)=0

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scisser | Student

Graph

Minimum at `-oo` and maximum at `oo`