Concavity_Presentation_Updated.pptx rana

For knowledge

1. Concavity in Mathematics Definition, Types, Graphs and Examples

2. Definition of Concavity Concavity describes the curvature of a graph. A function is concave up if it curves upwards, and concave down if it curves downwards.

3. Types of Concavity 1. Concave Up: The graph opens upward like a cup. 2. Concave Down: The graph opens downward like a frown. Concavity is determined using the second derivative of a function.

4. Concave Up A function is concave up on an interval if its second derivative is positive on that interval.

5. Concave Down A function is concave down on an interval if its second derivative is negative on that interval.

6. Graph: Concave Up and Down

7. Point of Inflexion A point of inflexion is a point on the graph where the concavity changes. At this point, the second derivative changes sign.

8. Graph: Point of Inflexion

9. Example 1 Determine the concavity of f(x) = x^2. Solution:
First derivative: f'(x) = 2x
Second derivative: f''(x) = 2 > 0
So, f(x) is concave up everywhere.

10. Example 2 Determine the concavity of f(x) = -x^2. Solution:
First derivative: f'(x) = -2x
Second derivative: f''(x) = -2 < 0
So, f(x) is concave down everywhere.

11. Example 3 Find the point of inflection of f(x) = x^3. Solution:
First derivative: f'(x) = 3x^2
Second derivative: f''(x) = 6x
Set f''(x) = 0 => x = 0
So, point of inflection at x = 0.

12. Example 4 Determine the concavity of f(x) = ln(x). Solution:
First derivative: f'(x) = 1/x
Second derivative: f''(x) = -1/x^2 < 0 for x > 0
So, f(x) is concave down for x > 0.

13. Example 5 Determine concavity of f(x) = e^x. Solution:
First derivative: f'(x) = e^x
Second derivative: f''(x) = e^x > 0
So, f(x) is concave up for all x.

14. Example 6 Find intervals where f(x) = sin(x) is concave up or down. Solution:
Second derivative: f''(x) = -sin(x)
Concave up when -sin(x) > 0 => sin(x) < 0
Concave down when sin(x) > 0

15. Example 7 Determine the concavity of f(x) = 1/x on x > 0. Solution:
First derivative: f'(x) = -1/x^2
Second derivative: f''(x) = 2/x^3 > 0 for x > 0
So, concave up on x > 0.

16. Example 8 Determine concavity of f(x) = sqrt(x). Solution:
First derivative: f'(x) = 1/(2sqrt(x))
Second derivative: f''(x) = -1/(4x^(3/2)) < 0 for x > 0
So, concave down for x > 0.