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# Is f(x)=-x^5-2x^4+8x^2-4x+2 concave or convex at x=-5?

Feb 23, 2016

convex at x = -5

#### Explanation:

To test if a function is concave / convex at f(a) , require to find the value of f''(a).

• If f''(a) > 0 then f(x) is convex at x = a

• If f''(a) < 0 then f(x) is concave at x = a

hence $f \left(x\right) = - {x}^{5} - 2 {x}^{4} + 8 {x}^{2} - 4 x + 2$

$f ' \left(x\right) = - 5 {x}^{4} - 8 {x}^{3} + 16 x - 4$

and $f ' ' \left(x\right) = - 20 {x}^{3} - 24 {x}^{2} + 16$

$\Rightarrow f \left(- 5\right) - - 20 {\left(- 5\right)}^{3} - 24 {\left(- 5\right)}^{2} + 16 = 1916$

since f''(-5) > 0 then f(x) is convex at x = -5