For what values of x is $f (x) = x^{4} - x^{3} + 7 x^{2} - 2$ $f(x)=x^4-x^3+7x^2-2$ concave or convex?

Question

1561 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-03T12:15:02

Concave for all real numbers

Find $f''(x)$ $"f''(x)"$
When $f''(x) \geq 0$ $"f''(x) ≥ 0"$ , f(x) is Concave upward
When $f''(x) \leq 0$ $"f''(x) ≤ 0"$ , f(x) is Concave Downward

$f (x) = x^{4} - x^{3} + 7 x^{2} - 2$ $f(x)=x^4−x^3+7x^2−2$
$f'(x) = 4x^3 - 3x^2 + 14x$
$"f''"(x)= 12x^2 - 6x + 14$

$"f''"(x) ≥ 0$
$12x^2 - 6x + 14 ≥ 0$
This is true for all real numbers => f(x) is concave upward for R.

For what values of x is f(x)=x^4-x^3+7x^2-2f(x)=x4−x3+7x2−2f(x)=x^4-x^3+7x^2-2 concave or convex?