The document discusses the signs of quadratic functions and how to determine if they are positive definite, negative definite, or indefinite based on the values of a, Δ, and k. An example problem demonstrates how to find the values of k that make the quadratic function kx2 - 6x + k positive definite by analyzing the signs of a and Δ. The solution is that k must be greater than 3.

1. The Sign Of A Quadratic

2. The Sign Of A Quadratic
Positive Definite
y
x

3. The Sign Of A Quadratic
Positive Definite
y
x

4. The Sign Of A Quadratic
Positive Definite
y
x
a0 , 0

5. The Sign Of A Quadratic
Positive Definite Negative Definite
y y
x x
a0 , 0

6. The Sign Of A Quadratic
Positive Definite Negative Definite
y y
x x
a0 , 0

7. The Sign Of A Quadratic
Positive Definite Negative Definite
y y
x x
a0 , 0 a0 , 0

8. Indefinite

9. Indefinite
y
x

10. Indefinite
y
x
a0 , 0

11. Indefinite
y y
x x
a0 , 0

12. Indefinite
y y
x x
a0 , 0 a0 , 0

13. Indefinite
y y
x x
a0 , 0 a0 , 0
y
x

14. Indefinite
y y
x x
a0 , 0 a0 , 0
y
x
a0 , 0

15. Indefinite
y y
x x
a0 , 0 a0 , 0
y y
x x
a0 , 0

16. Indefinite
y y
x x
a0 , 0 a0 , 0
y y
x x
a0 , 0 a0 , 0

17. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite

18. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0

19. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0
k 0

20. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0 0
k 0

21. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0 0
k 0 36  4k 2  0

22. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0 0
k 0 36  4k 2  0
k2  9

23. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0 0
k 0 36  4k 2  0
k2  9
k  3 or k  3

24. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0 0
k 0 36  4k 2  0
k2  9
k  3 or k  3
k  3

25. e.g. Find the values of k which makes kx 2  6 x  k  0 positive definite
a0 0
k 0 36  4k 2  0
k2  9
k  3 or k  3
k  3
Exercise 8G; 2ace, 3bd, 4bd, 5bd, 6, 12, 15, 17*