WEBVTT
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we're asked to find the minimum vertical distance between the
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parade. Bolas Y equals X squared plus one and
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y equals X minus X squared. Okay, big
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bar. Well, to do this first nous the
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vertical distance between these two dfx. This is the
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absolute value of X squared plus one minus X minus
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X squared, which is the absolute value of two
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X squared minus X plus one. You know,
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it's like now we know that the distance G is
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well minimized. Black when it's square d squared,
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which I'll call f is minimized as well. Now
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the function f of X, this is D squared
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of X, which is no, it's a piece
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of shit. I could have two x squared minus
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X plus one squared Mhm. All right. Now
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, in order to find the minimum vertical the minimum
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value of f obey dot com I'm going to take
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the derivative of F. So we have F Private
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X is by the chain rule two times two x
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squared minus X plus one times four x minus one
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So here is equal to zero. So the critical
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values satisfy two X squared minus X plus one equals
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zero or four X minus one equals zero. Now
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the discriminate of this quadratic equation Delta, this is
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B Square, which is one minus four times a
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, which is two times See, So this is
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one minus eight, which is negative seven, which
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is less than zero. So this first equation has
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no real solutions. The second equation has the real
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solution. X equals 1/4. Now, at this
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point, we're only considering Nick convinced himself. We
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haven't restrained our function to an interval, however,
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noticed that this our function f of X is in
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fact approaches positive infinity as X approaches plus or minus
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infinity. Therefore, it follows that F has an
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absolute minimum on its domain, and it's absolute minimum
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must occur at one of the critical values. So
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it follows that F has absolute quite sure minimum at
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X equals 1/4 and therefore, yeah, the minimum
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vertical distance is D of 1/4 which is the absolute
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value, and I'm of two times when fourth squared
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is 1/16 minus 1/4 plus one. This is the
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same as 1/8 minus 1/4 plus one. This is
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1/8 minus 2/8 plus 8/8 which is nine minus two
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is 7/8 absolute value of 7/8 which is just 7/8
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. What? This is our answer.