How do i do question 4 im confused by it asking to prove...

How do I do question 4? I'm confused by it asking to prove with a drawn graph. Suppose that f is a continuous function on the interval a, b and is symmetric about the point (the point whose x-coordinate is the midpoint of a, b and whose y-coordinate is the corresponding point on the graph of f). Using Step 2, show that f satisfies f(x) + f(a + b - x) = 2f((a + b)/2) for all x in a, b. Use Step 3 to show that a function that is continuous on a, b and is symmetric about the point ((a + b)/2, f((a + b)/2)) can be integrated as follows: ∫_a^b f(x)dx = (b - a)f((a + b)/2) = (1/2)(b - a)(f(a) + f(b)) For now, this proof can be done by drawing a careful picture and using the symmetry of the graph.

Transcript

00:01 Okay, in this question, we are going to discuss this function, fx being x squared.

00:07 Okay, we want to consider its concavity.

00:12 Okay, we'll show this concave.

00:17 We'll concave up.

00:19 Okay, in the graph, this concave up property is very easy because it's not very difficult for us to get the graph for this function.

00:32 It's a parabolo, right? so the graph would be like that.

00:38 Here is our x, here is our y, this is our original point 0, 0.

00:43 Okay, we say this function is concave up if and only if.

00:49 Let's choose any two points.

00:51 Let's say this is p, this is q.

00:55 Okay, the coordinate for p, let's assume it is a, a squared.

00:59 And the coordinate q is b squared.

01:03 We say this function is concave if the line segment of this straight line, i say p and q, always lies above the graph of this function.

01:17 But here we need to notice, we see the graph means the graph determined by those two points.

01:23 I mean, it's straightforward if we want to consider those two parts, those two pieces then, and those two pieces just lie, both lie above this line segment.

01:35 That means given those two points, we say we have some parts of our parabola or some part of our graph determined by those two points.

01:47 I think this red part.

01:51 We say we define the concave -r property if this is true for any two -point.

02:01 Given any two -point, the green line segment always lies above the red one.

02:12 Once the graph of a function has this property, then we say it is concave -bound.

02:18 It is true for any p.

02:20 For example, we can choose, let's say p ' here and q ' here...

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