1. Second
Derivative 6A

2. Starter: use graph software to draw these graphs
Second derivative
KUS objectives
BAT find and use the second derivative
Starter: use graph software to draw these graphs
𝑦= 𝑥 3 + 3𝑥 2 +1
Can you see any connections between them?
𝑦= 3𝑥 2 +6x
𝑦=6𝑥+6

3. 𝐈𝐧𝐯𝐞𝐬𝐭𝐢𝐠𝐚𝐭𝐞 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭
Notes
Geogebra: second derivative
𝐈𝐧𝐯𝐞𝐬𝐭𝐢𝐠𝐚𝐭𝐞 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭
𝐨𝐟 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭

4. Goes from negative to positve
Notes
Geogebra: second derivative - + - + - +
Graph of f(x)
Local minimum - + +
Graph of f’(x)
Goes from negative to positve + + + + + +
Graph of f’’(x)
Gradient of the gradient
Is positive

5. To find the nature of a stationary point we can
Notes
To find the nature of a stationary point we can
Find where the gradient (first derivative) is zero
Find the second derivative
If the 2nd derivative is < 0 → MAXIMUM
If the 2nd derivative is > 0 → MINIMUM
The bit you are likely to find difficult is solving equations to find coordinates
Here are some tips:
Write any negative powers as fractions
Multiply through to get rid of fractions
Simplify
Factorise when possible

6. When x = 2, therefore (2, 3) is a local minimum
WB20 Find the coordinates of the stationary point on
the curve 𝑦=𝑥+ 4 𝑥 and determine its nature
When x = 2, therefore (2, 3) is a local minimum

7. When x = ¼ , therefore (¼ , -¼) is a local minimum
WB21 Find the coordinates of the stationary point on
the curve 𝑦=𝑥− 𝑥 and determine its nature
When x = ¼ , therefore (¼ , -¼) is a local minimum

8. Practice 1

9. Practice 1 solutions

10. Practice 2

11. Practice 2 solutions

12. Given that there is a stationary point where x=2, find the value of k
WB The curve 𝑦= 𝑥 3 −𝑘 𝑥 2 +2𝑥−5 has two stationary points
Find 𝑑𝑦 𝑑𝑥
Given that there is a stationary point where x=2, find the value of k
Determine whether this stationary point is a min or max
Find the x-coordinate of the other stationary point

13. One thing to improve is –
KUS objectives BAT find and use the second derivative
self-assess
One thing learned is –
One thing to improve is –

14. END