HOW TO FIND THE MINIMUM AND MAXIMUM POINT OF A CURVE

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PROCEDURES FOR FINDING TURNING POINTS (MINIMUM AND MAXIMUM POINT) OF A CURVE OR FUNCTION

1. Given a function $y=f(x)$

2. Differentiate the function and obtain $\frac{dy}{dx}$.

3. Equate $\frac{dy}{dx}$ to zero

4. To get the corresponding coordinate (y) insert the value (or values) of x obtained in step 3 to in the original function.

5. To determine the nature of the turning point, take the second derivative of the function and obtain $\frac{d^2y}{dx^2}$

6. Insert the value(s) of x in the second derivative 

7. Observe the sign after solving step 6. If it's:

A) positive, It is a minimum point.

B) Negative, it is a maximum point.

Example

Find the minimum and maximum point of the curve $y=x^3-3x^2-9x+3$

Solution

Step 1: Given y=f(x)

$y=x^3-3x^2-9x+3$

Step 2: Differentiate the function and obtain $\frac{dy}{dx}$.

$$\frac{dy}{dx}=3x^2-6x-9$$

Note: We use the power rule.

Step 3: Equate $\frac{dy}{dx}$ to zero

$$3x^2-6x-9=0$$

Here we have a quadratic equation, Hence we solve quadratically

$$3x^2-6x-9=0$$

For simplicity, we divide through by 3

$$\frac{3x^2}{3}-\frac{6x}{3}-\frac{9}{3}=\frac{0}{3}$$

$$x^2-2x-3=0$$

$$x^2-3x+x-3=0$$

$$x(x-3)+1(x-3)=0$$

$$x=3, x=-1$$

Step 4: To get the corresponding coordinates, insert the values of x obtain in step 3 to get the original function: $y=x^3-3x^2-9x+3$

When x=3

$$y=(3)^3-3(3)^2-9(3)+3$$

$$y=27-27-27+3$$

$$y=-24$$

When x=-1

$$y=(-1)^3-3(-1)^2-9(-1)+3$$

$$y=-1-3+9+3$$

$$y=8$$

Therefore, the turning points are (3, -24) and (-1,8).

Step 5: To determine the nature of the turning point (whether it is a minimum or maximum point), take the second derivative of the function and obtain $\frac{d^2y}{dx^2}$

If $\frac{dy}{dx}=3x^2-6x-9$, using the idea of successive differentiation, then

$$\frac{d^2y}{dx^2}=6x-6$$

Step 6: Insert the value(s) of x in the second derivative.

When x=3

$$6(3)-6=12$$

When x=-1

$$6(-1)-6=-12$$

Step 7: Observe the sign after solving step 6

A) Because 12 is positive, (3, -24) is the minimum point of the curve.

B) Because -12 is negative, (-1,8) is the maximum point of the curve.

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