5. (a) The graph of a function y = f(x) is shown. At which...

5. (a) The graph of a function y = f(x) is shown. At which point(s) are the following true? (i) dy/dx and d^2y/dx^2 are both positive. (ii) dy/dx and d^2y/dx^2 are both negative (iii) dy/dx is negative but d^2y/dx^2 is positive (b) The graph of the derivative of the function, f', is shown below. i. Find all values of x that f has a local minimum. ii. Find all values of x that f has the inflection point(s).

Transcript

00:01 All right, so we have a two -part question.

00:02 The first part says we're given the graph of y equals f of x.

00:08 Looks something like this.

00:14 We're told here's point a, point b, point c, d, and e.

00:29 And i want to know where the first derivative and the second derivative are both positive.

00:52 So if the first derivative is positive, that means we're increasing.

00:58 If the second derivative is positive, that means we're concave up.

01:03 Well, the graph is decreasing at a.

01:05 The graph is increasing at b and is concave up.

01:10 So b, the graph is increasing at c, but concave down.

01:17 It's, we have a horizontal tangent line at d, and it's decreasing at e.

01:23 So where the derivative and the second derivative are both positive is point b.

01:28 The next one, they want to know when the derivative and the second derivative are both negative.

01:55 And if the second derivative is negative, then we know that it is, the function is decreasing.

02:06 And if the second, sorry, if the first derivative is negative, it's decreasing.

02:10 And if the second derivative is negative, then it's concave down.

02:19 So let's go ahead and look at for those points, it's decreasing at a, but it's going to.

02:26 Concave up at a.

02:28 It's increasing at b, increasing at c, it's level at d.

02:33 We have horizontal tangent line, and it is decreasing at e, and it is concave down there.

02:38 So this is going to be e.

02:45 And the third criteria was the first derivative is negative, and the second derivative is positive.

03:15 Okay, so again, first derivative being negative is decreasing.

03:21 Second derivative being positive, it's concave up.

03:28 And at the point a, it is decreasing and concave up.

03:34 Point b, it's increasing c, increasing, d, no.

03:38 E is decreasing, but concave down.

03:41 So there's that.

03:43 So where the derivative is negative, it's, sorry, where the derivative is negative and the second derivative is positive is point a...

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