Everything Maths Grade 12

CHAPTER 7. DIFFERENTIAL CALCULUS 7.5

Example 10: Calculation of TurningPoints

QUESTION

Calculate the turning points of the graph of the function

f (x) = 2x^3 − 9 x^2 + 12x− 15

.

SOLUTION

Step 1 : Determine the derivative of f (x)

Using the rules of differentiation we get:

f�(x) = 6x^2 − 18 x + 12

Step 2 : Set f�(x) = 0 and calculate x-coordinate of turning point

6 x^2 − 18 x + 12 = 0

x^2 − 3 x + 2 = 0

(x− 2)(x− 1) = 0

Therefore, the turning points are at x = 2 and x = 1.

Step 3 : Substitute x-coordinate of turning point into f (x) to determine y-coordinates

f (2) = 2(2)^3 − 9(2)^2 + 12(2)− 15

= 16− 36 + 24− 15

=− 11.

f (1) = 2(1)^3 − 9(1)^2 + 12(1)− 15

= 2− 9 + 12− 15

=− 10

Step 4 : Write final answer

The turning points of the graph of f (x) = 2x^3 − 9 x^2 + 12x− 15 are (2;−11)

and (1;−10).

We are now ready to sketch graphs of functions.

Method:

Sketching Graphs: Suppose we are given that f (x) = ax^3 + bx^2 + cx + d, then there are five steps to
be followed to sketch the graph of the function:

1. Determine the valueof the y-intercept by substituting x = 0 into f (x)