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

##### 1 Answer
Dec 25, 2015

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

#### Explanation:

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

Step 1: Find the derivative

$f ' \left(x\right) = 4 {x}^{3} - 6 {x}^{2} - 9$

Step 2: Differentiate again with respect to x

${f}^{2} \left(x\right) = 12 {x}^{2} - 12 x$

Step 3: Substitute $x = - 1$ in ${f}^{2} \left(x\right)$ and check for sign

${f}^{2} \left(- 1\right) = 12 {\left(- 1\right)}^{2} - 12 \left(- 1\right)$
${f}^{2} \left(- 1\right) = 12 + 12$
${f}^{2} \left(- 1\right) = 24$

If ${f}^{2} \left(x\right) > 0$ then the curve is convex.
If ${f}^{2} \left(x\right) < 0$, 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

http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx