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Let \(a,b,\) and \(t\) be real numbers such that \(a + b = t.\) Find, in terms of \(t,\) the minimum value of \(a^2 + b^2.\)

 Apr 25, 2019

Best Answer 

 #1
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\(b=t-a\\ a^2+b^2 = a^2 + (t-a)^2 = \\ 2a^2 - 2at + t^2 = \\ 2\left(a-\dfrac t 2 \right)^2 + \dfrac{t^2}{2}\\ \text{This clearly has a minimum value of }\dfrac{t^2}{2} \text{ at }a = \dfrac t 2\)

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 Apr 26, 2019
 #1
avatar+6046 
+1
Best Answer

\(b=t-a\\ a^2+b^2 = a^2 + (t-a)^2 = \\ 2a^2 - 2at + t^2 = \\ 2\left(a-\dfrac t 2 \right)^2 + \dfrac{t^2}{2}\\ \text{This clearly has a minimum value of }\dfrac{t^2}{2} \text{ at }a = \dfrac t 2\)

Rom Apr 26, 2019

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