# For what values of x is #f(x)=(x-2)(x-7)(x-3)# concave or convex?

##### 1 Answer

Concave on

#### Explanation:

The concavity and convexity of a function are determined by the sign (positive/negative) of the **second derivative**.

- If
#f''(a)<0# , then#f(x)# is concave at#x=a# . - If
#f''(a)>0# , then#f(x)# is convex at#x=a# .

In order to find the second derivative, we should first simplify the undifferentiated function by distributing.

#f(x)=(x^2-9x+14)(x-3)=x^3-12x^2+41x-42#

Now, find the first and second derivatives through a simple application of the power rule.

#f'(x)=3x^2-24x+41#

#f''(x)=6x-24#

Now, we must find the times when

#6x-24=0#

#6x=24#

#x=4#

The sign of the second derivative, and by extension, the concavity/convexity of the function, could shift only at

**When #mathbf(x <4)#:**

Test point at

#f''(0)=6(0)-24=-24#

Since this is

**When #mathbf(x >4)#:**

Test point at

#f''(5)=6(5)-24=6#

Since this is

We can check the graph of

graph{x^3-12x^2+41x-42 [-2, 10, -30, 15]}