Do convex functions have only one minimizer?

Question:

Do convex functions have only one minimizer?

Concavity:

For a well-defined function in an interval i.e. being continuous and differentiable everywhere, if the second-order derivative is positive {eq}f''(x)>0 {/eq} then it is said to be convex as well as if the second-order derivative is negative then it is said to be concave or concave downwards i.e. {eq}f''(x)<0. {/eq}

Answer and Explanation:


As by the definition of convexity, we know that - for a well-defined function in an interval i.e. being continuous and differentiable everywhere, if the second-order derivative is positive {eq}f''(x)>0 {/eq} then it is said to be convex. Now in addition:

  • If we have {eq}f'(x)=0 {/eq} once happening within the interval as happening for more than one time - it shall challenge the concavity{eq}\Rightarrow {/eq} we will be having concave upward graph with a single turn i.e. max to max only one minimum value or minimizer.

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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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