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# help

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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

#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

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