My teacher has given some test prep/homework questions. She says fully analyze them which means find the intercepts, symmetry, asymptotes, intervals of inc/dec, local max and min, inflection points, and concavity. Here are a couple of questions from the sheet. The first one I've gotten part way through (I put up my rough copy of it) and the second one I can't get because of the double root.
Here's my rough copy of the attempt at solution for the first one but I know that parts of it are wrong:
I got a couple of questios. For the first one (#17):
Is there no point of inflection?
And when I plugged in -1, I get f(-x)= f(x) therefore is the function not symmetric about the x-axis instead of origin?
For the 2nd one (#21), I see that there isn't a point of inflection but is it symmetric about the origin or x-axis?
now f(-x) = (-x) - 3(-x)^(1/3) = -x + 3x^(1/3) = -f(x)
since f(-x) = -f(x), we are symmetric about the x-axis and not symmetric about the y-axis and origin
see Reflections and Symmetry it tells you the different tests for symmetry, you can check them yourself. there is a pink table in the middle of the page entitled "Types of symmetry" under the sub-heading "Ways to test different symmetry" in big bold blue letters, use this table to answer your questions about 21, if you have problems, say so