HARD graph questions

[tolga]

Q)Consider the function f:R goes to R, f(x)=(x-1)^2(x-2)+1
a)the coordinates of the turning points of the graph of y=f(x) are (a,1) and (b,25/27). Find the values of a and b.
b)Find the real value of p for which the equation f(x)=p has exactly one solution.
c)For the following, k is a positive real number.
i)Describe a sequence of transformations that maps the graph of y=f(x) onto the graph of y=f(x/k)-1.
ii)Find the x-axis intercepts of the graph of y=f(x/k)-1 in terms of k.
d)Find the real values of h for which only one of the solutions of the equation f(x=h)=1

Q)Given two functions:
f:[0,6]goes to R, f(x)=1/4x(x-6)^2
g:[0,6]goes to R, g(x)=-x^2 =6x
Find the maximum value of {f(x)-g(x)} for 0<x<6, corect to two decimal places, and determine the values of x in this interval for  which the maximum occurs.

[andy456]

For the first question part a. I advise you draw the graph, that way it is easy to determine the value of a. As for b the book has misprinted its meant to be (b,23/27). Then sub it in to solve for b

[matrix]

first draw the graph of y=(x-1)^2(x-2) ad then move it 1 unit up. you can see what 'a' is equal to.
to find 'b' derive f(x) which should give you x coordinates of TP.

for part b) draw a line y=p on the same set of axis. where does it cut the graph of f(x) only once?

for part c) is it the same function f(x)=(x-1)^2(x-2)+1 or a general case?
dilation factor of k parellel to x axis, and then translated 1 down. (sub x with x/k)