Sketch the Graph

Question

Sketch the graph of f(x) = x^4 - 4x^3 + 1

Please show all work

Answer 1

The domain is all reals obviously.

P = {+-1}

Q = {+-1}

P/Q = {+-1}=P=Q are the only POSSIBLE rational solutions.

f(1) = 1 - 4 + 1 = -2 and f(-1) = 1 + 4 + 1 =6

so there are NO rational solutions.

We must find the solutions numerically, or by looking at the graph which we are asked to produce.

Note that f(0) = 1 while f(1) = -2. So there is a solution in the interval [0,1]

Also f(3) = 81 - 108 + 1 = -26 while f(4) = 256 - 256+1 = 1.

So there is another solution in the interval [3,4]

Since there are 2 sign changes, there are two positive real solutions,

we have found to be in between X1 in [0,1] and X2 in [3,4]

replacing x with -x, the function becomes f(x) = x^4 + 4x^3 + 1 which is positive everywhere.

SO there are no negative solutions.

Therefore, we have two imaginary solutions which we can find once we numerically

determine the two irrational positive solutions.

Sign chart shows the function is positive for x < X1, negative for X1 < x < X2 and

positive for x>X2

Using the Bisection method or Newton's method, X1=0.67 and X=3.984 APPROXIMATELY.

You can graph functions at desmos dot com