Extrema of Expressions, and Counting Roots

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. What is the greatest value of three sine x plus four cos x? Easy, right? Sine x is at most one, cos x is at most one, so three plus four: seven! Is that really true? Hold that thought. Today: quick tools for greatest and least values, and for counting the real roots of a cubic. They come up often in JEE problems. So, let's begin.

0:30Sine and cosine together

So, back to our puzzle: three sine x plus four cos x is at most three plus four, seven. The sign, less than with a line under it, means less than or equal to, and the question mark means we are not sure yet. Now tell me: can sine x and cos x both equal one at the same x? No, never! Sine x is one at x equals pi over two; pi is the Greek letter you see. But right there, cos x is zero. So the two terms never peak together, and seven is never reached. Oh no, careful: that is the trap! Right? So here is the real tool. As x runs over all real numbers, a sine x plus b cos x takes every value from minus root of a squared plus b squared, to plus root of a squared plus b squared. The tick sign is the square root, the small two means squared, and the square brackets mean both ends are reached. Clear? Now our puzzle: a is three, b is four. Root of nine plus sixteen is root twenty five: five. So the maximum is five, not seven! And the minimum is minus five. With me?

1:41AM ≥ GM: quick minimum values

Second tool, for minimum values: the A M and G M inequality. For u and v greater than zero, u plus v is greater than or equal to two root u v; that sign, greater than with a line under it, means greater than or equal to. In brackets, A M greater than or equal to G M: the arithmetic mean is at least the geometric mean. Okay? Now think. Suppose the product, u times v, is a fixed number. What does this line tell you about the sum, u plus v? Yes, exactly! The sum can never go below two times the root of that constant: a fixed lower bound. Got it? But careful. That bound is the minimum only if it is actually reached, and that happens exactly when u equals v. Always check this! Remember our seven? Same story. Clear? Here is a form that comes up often. For a and b greater than zero, a sec squared theta plus b cosec squared theta is at least root a plus root b, the whole squared. Theta is the Greek letter here. We will see where this comes from in Example one. With me?

2:50Counting the real roots of a cubic

Third tool: counting roots. The graph is y equals x cubed minus three x; the small three means cubed. The orange dot is a local max at x equals minus one, height two. The green dot is a local min at x equals one, height minus two. The dashed lines are flat tangents: the note at the top says f prime, the derivative, is zero at both; the little dash is read prime. We call these local extreme values capital M and small m, okay? Now count. How many times does this curve cross the x axis? Three times. Correct! The hilltop is above the axis and the valley is below it, so the curve must cross before, between and after them. So the rule: a cubic has three distinct real roots if and only if M times m is negative; that double-headed arrow means if and only if. Right? Here, capital M is two and small m is minus two: the product is minus four, less than zero. So, three distinct real roots, just as the picture shows. Got it?

4:00Extrema given: set f'=0 there

Last tool. Extrema means the local maximum and minimum points. If a differentiable function has extrema at x equals p and x equals q, then f prime is zero at both; the double arrow means implies. Every polynomial is differentiable. Clear? Quick course example: f of x equals x cubed plus a x squared plus b x has extrema at x equals one and three. Find a and b. f prime of x is three x squared plus two a x plus b. So, which two equations do we get? f prime of one is zero: three plus two a plus b equals zero. f prime of three is zero: twenty seven plus six a plus b equals zero. Solve them: a is minus six, b is nine. With me?

4:54Example 1: minimum of 9 sec squared + 16 cosec squared

Now, Example one. Find the minimum of nine sec squared theta, plus sixteen cosec squared theta. Pause the video, and try it first. Done? Okay, let's check. Some students say: sec squared theta is at least one, cosec squared theta is at least one, so the minimum is nine plus sixteen, twenty five. Is that right? No, no, wait! Same trap as our seven: the two terms are never both one at the same theta. So, write sec squared as one plus tan squared, and cosec squared as one plus cot squared. Okay? Nine plus sixteen gives twenty five, plus nine tan squared theta plus sixteen cot squared theta, both positive. Now, what is the product of these two terms? One hundred and forty four. Perfect! Since tan squared times cot squared is one, the product is a constant. So by A M and G M, their sum is at least two root of nine times sixteen; the dot means times. That is two times twelve: twenty four. Right? Equality holds when nine tan squared theta equals sixteen cot squared theta, that is, tan squared theta is four over three. Such a theta exists, so twenty four is reached. Clear? So the minimum is twenty five plus twenty four: forty nine. And forty nine is three plus four, squared: root nine plus root sixteen, squared, our formula from before. Lovely!

6:31Example 2: k for 3 distinct real roots of x cubed - 3x + k

Now, Example two, a little harder. For which k does x cubed minus three x plus k equal zero have three distinct real roots? Pause the video, and try it. The figure helps. Done? Okay, let's check. f prime of x is three x squared minus three, zero at x equals plus or minus one; that is the plus minus sign. The same points as in the figure! Adding k only lifts the graph up or down. With me? So the local max is f of minus one, which is k plus two. And the local min is f of one, which is k minus two. Clear? Now use the rule. For which values of k is capital M times small m negative? k plus two, times k minus two, must be negative: opposite signs. That happens exactly when k lies strictly between minus two and two. In the picture: hilltop above the axis, valley below. Got it? Oh no, careful at the ends! If k is two or minus two, M times m is zero: the curve just touches the axis, giving a repeated root, not three distinct roots. So, minus two less than k less than two, strictly. Right?

7:53Quick revision: sin-cos range, AM-GM, roots of a cubic

So, quick revision. The expression a sine x plus b cos x runs between minus and plus root of a squared plus b squared. Next, A M is at least G M for positive terms, but check that equality can hold. Third, a cubic has three distinct real roots exactly when M times m is negative. And extrema at p and q give f prime zero at both. Now go and try the practice problems on this page. All the best!