Is $f (x) = x^{4} - 2 x^{3} - 9 x - 14$ $f(x)=x^4-2x^3-9x-14$ concave or convex at $x = - 1$ $x=-1$ ?

Question

1987 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-25T09:15:26

At $x = - 1$ $x=-1$ the curve is Convex or Concave Up Explanation given below.

$f (x) = x^{4} - 2 x^{3} - 9 x - 14$ $f(x) = x^4-2x^3-9x-14$

Step 1: Find the derivative

$f'(x) = 4x^3-6x^2-9$

Step 2: Differentiate again with respect to x

$f^2(x) =12x^2-12x$

Step 3: Substitute $x=-1$ in $f^2(x)$ and check for sign

$f^2(-1)=12(-1)^2-12(-1)$
$f^2(-1)=12+12$
$f^2(-1)=24$

If $f^2(x) >0$ then the curve is convex.
If $f^2(x) <0$ , then the curve is *concave *

We can see at $x=-1$ the second derivative is greater than zero, hence, the curve is convex.

For further information, you can refer
Note: Concave up is same as convex and concave down is concave

Is f(x)=x^4-2x^3-9x-14f(x)=x4−2x3−9x−14f(x)=x^4-2x^3-9x-14 concave or convex at x=-1x=−1x=-1?