Increasing and Decreasing Functions

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Take two numbers: e to the power pi, and pi to the power e. Here e is about two point seven, and pi is about three point one four. No calculator. Which one is bigger? Hard to say, right? The base and the power just swap places! Today, one simple idea, the sign of the derivative, will settle it. And this idea comes up often in JEE: rising and falling, inequalities, and counting roots. So, let's begin.

0:35The sign of f'(x) decides

So, first the heading: f prime of x. The little dash is called prime; f prime of x is the derivative, the slope of the tangent. Now think. If the slope is positive at every point, what does the graph do as you move to the right? It goes up! Yes, exactly. So, if f prime of x is greater than zero on an interval, then f is strictly increasing there: as x increases, f of x increases. The double arrow means implies. And note: on an interval, so f prime must exist and stay positive at every point of it. Okay? The same way, if f prime of x is less than zero on an interval, then f is strictly decreasing there: the graph keeps going down. Simple. Right? Now a tricky one. Suppose f prime of x is zero at just one point, and positive everywhere else. Is f still strictly increasing? Yes, it is! Careful here. If f prime of x is greater than or equal to zero, that is the sign with the line below it, and it equals zero only at isolated points, single separate points, not a whole stretch, then f is still strictly increasing. The tangent is flat for an instant, and the graph keeps climbing. Got it?

1:50Finding the intervals

Now, how do we find where a function increases, and where it decreases? Three steps. Step one: solve f prime of x equals zero. At these points, the sign of f prime may change. With me? But wait. Is that the only place where f prime of x can change its sign? No! The sign can also change at a point where f prime of x is undefined. So, step two: find those points too. Clear? Step three: test the sign of f prime of x between these points. Just put any one value of x from each gap into f prime. Positive means increasing, negative means decreasing. Okay?

2:33Proving inequalities

Alright, now a powerful use: proving inequalities. To prove f of x is greater than g of x, for x greater than a, take a new function: h equals f minus g. Right? Then show two things: h of a is greater than or equal to zero, and h prime of x is greater than zero for x greater than a. So h starts at zero or above, and keeps climbing: it never drops below zero! Okay? Let us try it on a course example. Prove that sine x is less than x, for x greater than zero. Take h of x equals x minus sine x. At zero, h is zero minus sine zero, which is zero. Clear? Now, what is h prime of x? And is it ever zero? h prime of x is one minus cos x. Cos x is never more than one, so this is never negative. But it is zero whenever cos x is one: at two pi, four pi, and so on; pi is the Greek letter, and the three dots mean and so on. Single, separate points: isolated! With me? So, by the isolated points rule, h is strictly increasing. For x greater than zero, h of x is greater than h of zero, which is zero. So sine x is less than x. Lovely! Got it?

3:54Counting roots

One more use, and then the examples. Can a strictly increasing function cross the x axis twice? No! To cross it again, the graph would have to come back down, and it never does. So, a strictly monotonic function, that is, one that is only increasing or only decreasing, has at most one root in an interval. At most one: maybe one, maybe none. Okay?

4:21Example 1: where x3 - 6x2 + 9x + 1 increases and decreases

Now, Example one. Find where f of x equals x cubed, minus six x squared, plus nine x, plus one, is increasing, and where it is decreasing. Pause the video, and try it with the three steps. Done? Okay, let's check. Differentiate: f prime of x equals three x squared, minus twelve x, plus nine. Take three common and factorise: three times x minus one, times x minus three. Right? Step one: f prime of x is zero at x equals one, and at x equals three. Step two: f prime is a polynomial, so it is defined for all x. No extra points. Clear? Now, the sign between one and three. The leading number, three, is positive, so f prime is positive there too, right? No, no, careful! Never guess the sign; test it. At x equals zero, f prime is nine: positive. At x equals two, it is minus three: negative! At x equals four, it is nine again: positive. So the sign goes plus, minus, plus. With me? So, f is increasing on minus infinity to one, and on three to infinity, and decreasing on one to three. The sideways eight is infinity, and the round brackets mean open intervals: the end points are not included. That's it!

5:48Example 2: which is larger, e to the pi or pi to the e?

And now Example two: our puzzle from the start. Which is larger, e to the power pi, or pi to the power e? Pause the video, and try it. Hint: the function log x over x. The l n you see is log, the natural log, to the base e. Done? First, a quick guess. Pi is bigger than e, so pi to the power e should be bigger, right? Hmm, careful! Guessing is not proving. Let us use monotonicity instead. Take f of x equals log x over x, for x greater than zero. Now, find f prime of x with the quotient rule. f prime of x equals one minus log x, over x squared. Correct! Now, x squared is positive. And for x greater than e, log x is greater than one, so the top is negative. So f prime is negative, and f is strictly decreasing for x greater than e. Right? Now, pi is greater than e, and f is decreasing there. So f of pi is less than f of e. Log e is one, so f of e is one over e. That gives: log pi over pi is less than one over e. With me? Multiply both sides by e times pi. It is positive, so the sign stays the same: e log pi is less than pi. Now, e log pi is log of pi to the power e, and pi is log of e to the power pi. Okay? And log is an increasing function, so the bigger log belongs to the bigger number. Pi to the power e is less than e to the power pi. So e to the power pi is larger! Our quick guess was wrong: the bigger base did not win.

7:33Quick revision: f prime sign, intervals, inequalities, roots

Got it? So, quick revision. f prime positive: increasing. f prime negative: decreasing. Isolated zeros of f prime are fine. For intervals: solve f prime equals zero, find where f prime is undefined, and test the signs. For inequalities: h equals f minus g, with h of a at least zero and h prime positive. And a strictly monotonic function has at most one root. Now go and try the practice problems on this page. All the best!