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In this video, we're going to look at how to find the absolute min and the absolute max of a continuous function on a closed interval.
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So here we have the continuous function g of x is equal to the absolute value of x plus 4 on the closed interval from negative 7 to 1.
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Now, the absolute value of x plus 4, its domain is all real numbers when i don't have a restriction on the interval that we're looking at.
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So there are no values of x for which that function is undefined.
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But here we want to focus specifically on the closed interval from negative 1 to 7.
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Now to do this, we first find the critical values of our function that are on that interval.
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So we want to find the critical values of g of x is equal to the absolute value of x plus 4 on the closed interval from negative 1 to 7.
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Then the second thing we're going to do is evaluate our function at the critical value or values, depending on what we have, found in step one.
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And then in step three, we want to evaluate the absolute g of x equals the absolute value of x plus four at the end points of our closed interval.
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So at negative seven, the left end point, and at one, the right end point.
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And then finally in step four, we're going to compare the outputs in steps two and three.
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The smallest is the absolute min and the largest is the absolute max.
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So the outputs are your extreme values and then the inputs are where they occur, what input values they occur.
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So finding the critical values of g of x is equal to the absolute value of x plus four.
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To do that, we want to look at rewriting the original function g of x as how we could rewrite it as a piecewise defined function and still mean the same thing as the absolute value of x.
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Plus 4.
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Well remember absolute value of a number is the number itself for numbers that are zero or greater and they're the opposite of the number for values that are negative.
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So i want to look at this x plus 4 and think what values of x would make x plus 4 0 or positive.
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So you kind of are thinking when is x plus four greater than or equal to zero we'll subtract the four over for x's that are greater than or equal to negative four now i'm going to put that on my second line of my piecewise defined function and if that's when the in the quantity that i'm taking the absolute value of is positive the absolute value of a positive or zero value is itself so i'm just going to write the x plus 4 and that's when the x is are greater than are equal to negative 4.
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Now, for values of x, for which what you're taking the absolute value of is less than 0, when is x plus 4 negative? when is it less than 0? subtract the 4 over when x is less than negative 4.
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I'm going to write that on the top line.
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And then when you're taking the absolute value of a quantity that's negative, you change.
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Its sign to give its absolute value so that's the opposite of x plus four and then we can kind of just think through pick a value like an x value that's less than negative four would be for example negative five well negative five plus four is negative one and the opposite of negative one is one so that's what you would have when you were looking at an absolute value of it now we're gonna rewrite this g of x function one more time because we want to get rid of the parentheses before we take the derivative.
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So it's negative x minus four for the xes that are less than negative four and it's x plus four for the x values that are greater than are equal to negative four.
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And the reason that i wrote them in this the top line and the bottom line this way, because i want to think of it from smaller values of x, from smaller values of x working up to larger values of x.
04:31
Okay, now what about the derivative? the derivative g prime of x i have to do in the piecewise defined way so the derivative of negative x is negative one and the derivative of the constant term negative four is zero so the derivative of negative x minus four is negative one and that happens for x that are smaller than negative four and then for values of x that are bigger than negative four greater than negative four the derivative of x is positive one and the derivative of four the term four is zero and now when i look at this as you get infinitely close to negative four from the left and the right your left hand limit is negative one your right hand limit is one and they don't coincide so this limit does not exist at x equal negative four so i don't have either of these lines that say or equal to so this derivative is undefined at x equal negative 4 and that's an x value for which the first derivative is undefined but that was a value in the domain of the original function and it's in our closed interval from negative 7 to 1 so that means that x equal negative 4 is a critical value for our function g of x and in fact it's a critical value we want to look at because it's in our closed interval now there are no values of x for which the first derivative is zero.
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It's either negative 1 or it's 1.
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So we don't have any values for which it's 0.
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Now go to step 2...