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03-10-2019
Mathematics

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Prove the inequalities. c+ 1/c ≥2 for c>0

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LammettHash LammettHash

03-10-2019

[tex]c+\dfrac1c=\dfrac{c^2+1}c=\dfrac{c^2-2c+1}c+2=\dfrac{(c-1)^2}c+2[/tex]

We have [tex](c-1)^2=0[/tex] when [tex]c=1[/tex], and for all other values of [tex]c[/tex] we have [tex](c-1)^2>0[/tex]. So

[tex]c+\dfrac1c=\dfrac{(c-1)^2}c+2\ge2[/tex]

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