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I have a function of $f(x)=x^5+x$ on the interval $[-2,2]$.

By using the first order derivative method of convexity we have $f(x) ≥f(x_0)+∇f(x_0)^T(x-x_0)$ and on the right hand side I get $-4x_0^5+5x_0^4+x$ from which we can't tell whether it's bigger or smaller than $x^5+x$.

I tried the second order derivative method as well, which states that $f(x)$ is convex if and only if $f''(x)≥0$ but the second order derivative of our function yields $20x^3$ which can be both negative and positive on our interval.

So does it mean that the function is neither convex nor concave?

Nick202
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    Yes this can certainly be the case, if the second derivative changes sign on your interval. –  Feb 03 '19 at 21:20
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    Of course, think about the sine function. – Did Feb 03 '19 at 21:20
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    You're right for this function-- neither concave nor convex on the entire interval. But it is one or the other on the positive or negative part of the interval. – coffeemath Feb 03 '19 at 21:21

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