Question

Describe the graph of the function f(x) = x3 − 11x2 + 36x − 36. Include the y-intercept, x-intercepts, and the shape of the graph.

Answer 1

Explanation:

y-intercept: x = 0, then y = f(0) = 0 - 0 + 0 - 36 = -36

x-intercept: y = 0 => factor the function (start by dividing by x -2)

f(x) = (x-2)(x-3)(x-6) =0 => x =2, x = 3, x = 6 (these are the x-intercepts)

critical points:

between x = 2 and x = 3, there is a local maximum

between x =3 and x = 6 there is a local minimum

Shape:

The function comes growing from - infinity.

In the third quadrant the function is negative (it does not pass through the second quadrant)

It enters to the fourth quadrant intercepting the y-axis at y = -36. It continues growing and intercepts the x-axis at x = 2.

It continues increasing until a maximum local positive value, starts to decrease, intercepts the x-axis at x = 3, continues decreasing, becomes negative, gets a local minimum in the fourth quadrant, starts to increase, intercepts the x-axis at x = 6, becomes positive, and continues growing.

Answer 2

Hello,

f(x)=x^3-11x²+36x-36
y-intercept
x=0==>y=-36
x-intercepts
f(2)=2^3-11*4+72-36=8-44+36=0

f(x)=(x-2)(x²-9x+18)
g(x)=x²-9x+18
g(3)=3²-9*3+18=0
g(x)=(x-3)(x-6)

f(x)=(x-2)(x-3)(x-6)



f'(x)=3x²-22x+36=(x-(11+√13)/3)(x-(11-√13)/3)
x≈4,8685... or x≈9,7981