7.5 CHAPTER 7. DIFFERENTIAL CALCULUS
Step 2 : Determine the y-intercept
We find the y-intercepts by finding the value for g(0).
g(x) =−x^3 + 6x^2 − 9 x + 4
yint= g(0) =−(0)^3 + 6(0)^2 − 9(0) + 4
= 4
Step 3 : Determine the x-intercepts
We find the x-intercepts by finding the points for which the function g(x) = 0.
g(x) =−x^3 + 6x^2 − 9 x + 4
Use the factor theoremto confirm that (x− 1) is a factor. If g(1) = 0, then
(x− 1) is a factor.
g(x) =−x^3 + 6x^2 − 9 x + 4
g(1) =−(1)^3 + 6(1)^2 − 9(1) + 4
=−1 + 6− 9 + 4
= 0
Therefore, (x− 1) is a factor.
If we divide g(x) by (x− 1) we are left with:
−x^2 + 5x− 4
This has factors
−(x− 4)(x− 1)
Therefore:
g(x) =−(x− 1)(x− 1)(x− 4)
The x-intercepts are: xint= 1; 4
Step 4 : Calculate the turning points
Find the turning points by setting g�(x) = 0.
If we use the rules of differentiation we get
g�(x) =− 3 x^2 + 12x− 9
g�(x) = 0
− 3 x^2 + 12x− 9 = 0
x^2 − 4 x + 3 = 0
(x− 3)(x− 1) = 0
clearpage The x-coordinates of the turning points are: x = 1 and x = 3.
The y-coordinates of the turning points are calculatedas:
g(x) =−x^3 + 6x^2 − 9 x + 4
g(1) =−(1)^3 + 6(1)^2 − 9(1) + 4
=−1 + 6− 9 + 4
= 0
