My girlfriend has a problem with her math task. I did all this stuff years ago when so I am pretty behind and clueless what to do. She has following function: $x^3 - 3x^2 +kx $$
Her tasks are following:
a) Analyze the function for $k = 1,$ $k = -1$ and $k = 0.$
b) Prove that all the functions $f_{k}$ have the same inflection point.
c) Analyze how $k$ influences the relative maxima and minima of $f_k.$
Note that one of the roots of this function is $0$, since we can factor out the $x$ this leaves us with $$x(x^2-3x+k)$$ the other roots can easily be found via discriminant.
The derrivative is $$3x^2-6x+k$$ the zeroes of it(which can be found via discriminant once again) correspond to the original functions relative maxima and minima.
The second derrivative is $$6x-6$$ since the free member $k$ become zero with derivation, and since the second derivatives zeroes are the inflection points of the function, we can say that the point of inflection is not affected by $k$.