Is $f (x) = - 4 x^{5} - 2 x^{4} - 2 x^{3} + 42 x - 30$ $f(x)=-4x^5-2x^4-2x^3+42x-30$ concave or convex at $x = 0$ $x=0$ ?

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Is $f (x) = - 4 x^{5} - 2 x^{4} - 2 x^{3} + 42 x - 30$ $f(x)=-4x^5-2x^4-2x^3+42x-30$ concave or convex at $x = 0$ $x=0$ ?

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Answer 1 · 2015-12-06T23:12:26

Neither of the two.

Explanation:

You need to check the second derivative: if it's positive in $x = 0$ $x=0$ , the function is convex. Otherwise, it's concave. So, we have

$f (x) = - 4 x^{5} - 2 x^{4} - 2 x^{3} + 42 x - 30$ $f(x)=-4x^5-2x^4-2x^3+42x-30$

$f'(x)=-20x^4-8x^3-6x^2+42$

$f''(x)=-80x^3-24x^2-12x$

Unfortunately, $f''(0)=0$ . Since it is positive before zero and negative after zero, it means that in $x=0$ the function passes from being convex to being concave.

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Is $f (x) = - 4 x^{5} - 2 x^{4} - 2 x^{3} + 42 x - 30$ $f(x)=-4x^5-2x^4-2x^3+42x-30$ concave or convex at $x = 0$ $x=0$ ?

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Is f(x)=-4x^5-2x^4-2x^3+42x-30f(x)=−4x5−2x4−2x3+42x−30f(x)=-4x^5-2x^4-2x^3+42x-30 concave or convex at x=0x=0x=0?

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Is $f (x) = - 4 x^{5} - 2 x^{4} - 2 x^{3} + 42 x - 30$ $f(x)=-4x^5-2x^4-2x^3+42x-30$ concave or convex at $x = 0$ $x=0$ ?