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I'm trying to do these problems, but I'm not sure how to start. Can someone help me figure out how I'm supposed to approach these? Thank you

Let a, b, c be positive constants. For all positive numbers x, y with product c, find the minimum value of ax + by.

If a, b, c are real numbers not all equal, prove that: a^2 + b^2 + c^2 > ab + ac + bc

Given any positive constant c, find the minimum value of x^4 + 2y^4 for positive numbers x and y having product xy = c.

user593248
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  • (i) Use the inequality between arithmetic mean and geometric mean, $(ax+by)/2\ge\sqrt{ax\cdot by}$. (ii) This is $(a-b)^2+(b-c)^2+(c-a)^2\ge 0$ rewritten is a simpler shape. Equality may occur! (iii) Use the same idea as in (i). Please use mathjax as here: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – dan_fulea Sep 14 '18 at 21:26

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You are given $xy=c$ and want to minimize $ax+by$. Solve the constraint for one of the variables and plug it into your objective, so here $y=\frac cx$ and the function to minimize is $ax+b\frac cx$. This is a function of one variable, so take the derivative and set it to zero.

For the second one subtract the right side from both sides making the right side zero. You should be able to express the new left side as a sum of squares. That says it is at least zero and strictly greater when one of them is not zero.

The third is similar to the first.

Ross Millikan
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