The definition of a convex function is as follows. The line segment connecting any two points on the graph is above the graph. But with this definition, I don't know why the convex function is U-shaped. I know that a U-shaped function satisfies this definition. but the converse. Why is the convex function U-shaped?
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Are you looking for an intuitive explanation or a detailed proof? If the latter, you will need to define precisely what you mean by "U-shaped". – David Apr 01 '22 at 05:46
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2Do you consider $f(x)=x$ as $U-$shaped? – Kavi Rama Murthy Apr 01 '22 at 05:47
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Yes. f(x) = |x| is also convex. – HackSMW Apr 01 '22 at 05:49
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@David I want an explanation. – HackSMW Apr 01 '22 at 05:51
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You have to explain what you mean by $U-$ shaped. As it stands the question does not make much sense (since you cannot prove the converse part or give a counter-example without knowing what $U-$ shaped means. – Kavi Rama Murthy Apr 01 '22 at 05:53
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I know that as the tangent slope of a differentiable convex function increases, it draws a U-shape. – HackSMW Apr 01 '22 at 05:55
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Wikipedia https://en.wikipedia.org/wiki/Convex_function In simple terms, a convex function refers to a function whose graph is shaped like a cup {\displaystyle \cup }\cup , while a concave function's graph is shaped like a cap {\displaystyle \cap }\cap . – HackSMW Apr 01 '22 at 05:57
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Given a sufficiently broad definition of what counts as "U-shaped", then it might be best to think in terms of "What happens if a function isn't U-shaped?".
If the function fails to be U-shaped, i.e. some of the function bends back on itself, then you can choose two points on either side of that bend and the line between them will have to cross over the bendy part of the function, which means that the function cannot be convex.
ConMan
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