Expert: Chen Min - 12/12/2008

Question
Sketch the following function without the use of a graphing calculator.  Show all work for and identify any x& y intercepts, asymptotes (if there are any), extrema, and points of inflection.

f(x)=4x^3-x^4
Thank you.

Answer
I cannot do the sketch for you, and you should do such basic job yourself.Besides, what is a graphing calculator?

When x = 0, y = 0 (0 is y-intercept and one of x-intercept)
when y = 0, x = 0 or x = 4, so 4 is another x-intercept

As f(x) and f(x) + x do not have any finite limit, there is no asymptotes.

f'(x) = 12x^2 - 4x^3
f''(x) = 24x - 12x^2
when x = 0 or x = 3, f'(x) = 0
since f''(x) is positive before and after x=0, (0,0)is neither a maximum nor a minimum, it is the point of inflection.
since f''(x) is positive before x=3, and negative after x=3, point(3,27) is a maximum.

Hope this helps.