Local Maxima and Minima

Transcript

0:00Introduction

Hello everyone! Picture a mountain road. At the top of a hill, the road is flat, right? And at the bottom of a valley, it is flat again. So, whenever the road is flat, must you be at a top or a bottom? Most students say yes. Today we see why the answer is no, and we even meet a bottom with no flat road at all, just a sharp corner! Questions on maxima and minima come up often in JEE, so let's begin.

0:29Hilltops and valleys: local maximum and minimum

So, on screen: y equals x cubed, minus three x. The small three up there means cubed. At x equals minus one, the curve climbs to a hilltop at height two: the orange dot, local max. At x equals one, it dips to a valley at minus two: the green dot, local min. With me? Now, the definition. f has a local maximum at c, if f of c is greater than or equal to f of x, for all x near c. The greater than sign with a line under it means greater than or equal to. Okay? Quick question. Is two, our hilltop, the largest value of the whole function? No! At the right end, the curve climbs higher than two. That is why we say local: f of c only has to beat the values near c. Similarly, f has a local minimum at c, if f of c is less than or equal to f of x, for all x near c. The less than sign with a line under it means less than or equal to. Clear? Now see the dashed lines at the hilltop and the valley: the tangents there, both flat. Flat means slope zero, so f prime is zero at both points, as the label says. The small dash, called prime, means the derivative. So, a critical point is a point of the domain where f prime of c is zero, or does not exist. Right? Each maximum or minimum is called an extremum. Key fact: every local extremum inside an interval is at a critical point. The double arrow means this gives that. Now the reverse. Is every critical point an extremum? Think of our flat road. No! The double arrow with a slash means does not give. Every critical point is only a candidate; we must test it, and we have two tests. Let's go.

2:27First derivative test: watch the sign of the slope

First test. Look at the hilltop at x equals minus one. Just to its left, is the curve rising or falling? And just to its right? Rising on the left, so f prime is positive; falling on the right, so negative. Yes, exactly! So, if f prime changes from plus to minus at c, f has a local maximum. The arrow means changes to. Okay? At the valley, x equals one, it is the other way round: minus to plus, so a local minimum. Got it? But if f prime keeps the same sign on both sides of c, there is no extremum at c, even if f prime of c is zero. Keep this in mind; a trap is coming. Clear?

3:14Second derivative test

Now the second test. f double prime, with two dashes, is the second derivative. If f prime of c is zero, and f double prime of c is less than zero, negative, we get a local maximum. But why? f double prime is the rate of change of f prime. So what does a negative value do to the slope? It makes the slope decrease! Correct! So f prime goes from plus, through zero, to minus: the hilltop of the first test. And if f double prime of c is greater than zero, the slope increases: a local minimum. With me? But careful! If f double prime of c is zero, the test fails; it decides nothing. Then go back to the sign of f prime. Okay?

4:01Two traps: flat but no extremum, extremum but not flat

Back to our mountain road. Take f of x equals x cubed: f prime of x is three x squared, so f prime of zero is zero. Flat road at zero! So, is zero a maximum or a minimum? Check the sign of f prime. Neither! Three x squared is never negative, so f prime is greater than or equal to zero on both sides. No change of sign, so no extremum. Flat road, but no top and no bottom! Got it? Now the other trap: f of x equals mod x. The two vertical bars mean the absolute value of x. Its graph has a sharp corner at zero, so f prime of zero does not exist. Right? And yet, mod x is never negative: it is greater than or equal to zero, which is f of zero. So zero is a minimum, with no derivative at all! That is why critical points include points where f prime does not exist. Puzzle solved. Clear?

5:01Example 1: local max and min of x cubed minus 3x plus 2

Now, Example one, an easy one. Find the local extrema of f of x equals x cubed, minus three x, plus two. Pause the video, and try it. Done? Okay, let's check. f prime of x is three x squared minus three, that is, three times, x squared minus one: zero at x equals plus or minus one. The plus over minus sign means plus or minus. Now, what is f double prime of x? Six x. Correct! The minus three is a constant, so its derivative is zero. Okay? At minus one, f double prime is six times minus one: minus six. Negative, so a local maximum. And f of minus one is minus one, plus three, plus two: four. Now x equals one. Maximum or minimum? Minimum! Very good. f double prime of one is six, positive. And f of one is one, minus three, plus two: zero. Right? Look at the figure again: y equals x cubed minus three x, our function without the plus two. So our curve is that curve, lifted up by two: the hilltop goes from two to four, the valley from minus two to zero. Lovely! With me?

6:27Example 2: x to the 4 minus 4x cubed, second test fails at 0

Now a harder one. Find the local extrema of f of x equals x to the power four, minus four x cubed. Pause the video, and check every critical point carefully. Done? Okay, let's check. f prime of x is four x cubed minus twelve x squared, that is, four x squared, times x minus three. So it is zero at x equals zero, and at x equals three. Clear? Two critical points, so two extrema, right? No, no, wait! Test them first. f double prime of x is twelve x squared minus twenty four x, which is zero at zero. So the second derivative test fails, and we go back to the sign of f prime. With me? Near zero, take x not equal to zero; the equals sign with a slash means not equal to. Then four x squared is positive and x minus three is negative, so f prime is negative on both sides. So, maximum or minimum at zero? Neither! No change of sign, so no extremum at zero. The curve flattens for a moment and keeps going down: an inflection point. Our mountain road trap again! Got it? Now, x equals three. Just before three, x minus three is negative, so f prime is negative; just after, positive. Minus to plus: a local minimum. f of three is eighty one, minus one hundred and eight: minus twenty seven. So, only one local extremum: a minimum of minus twenty seven, at three. Perfect!

8:11Quick revision: critical points and both derivative tests

So, quick revision. Critical points: f prime is zero, or does not exist. First test: f prime from plus to minus, a maximum; minus to plus, a minimum; no change, neither. Second test, when f prime of c is zero: f double prime negative, a maximum; positive, a minimum; zero, the test fails. Now go and try the practice problems on this page. All the best!