f(x)=(x-1)*(x+2)^3. Verify if the function is convex or concave.

If a function is convex, then the second derivative is
positive:

f"(x)>0

If a
function is concave, then the second derivative is
negative:

f"(x)<0

To
determine if the second derivative is negative or positive, we'll have to calculate, for
the beginning, the first derivative:

f'(x) =
[(x-1)*(x+2)^3]'

Since the function is a product, we'll
differentiate using the rule of product:

(u*v)' = u'*v +
u*v'

We'll put u = x-1

u' =
(x-1)'

u' = 1

v =
(x+2)^3

v' =
3*[(x+2)^2]*(x+2)'

v' =
3*[(x+2)^2]

f'(x) = (x+2)^3 +
3(x-1)*[(x+2)^2]

f"(x) = 3(x+2)^2 + 3(x+2)^2 +
3(x-1)*2(x+2)

f"(x) = 6(x+2)^2 +
6(x+2)(x-1)

We'll factorize by
6(x+2):

f"(x)
= 6(x+2)(x+2+x-1)

f"(x)
= 6(x+2)(x+1)

Now, we'll put f"(x) =
0:

6(x+2)(x+1) = 0

We'll put x
+ 2 = 0

x1 = -2

x+1 =
0

x2 =
-1

Between x = -2 and x = -1, f"(x) is
negative, and between (-infinite;-2) and (-1;+infinite), f"(x) is
positive.

The function is
convex for x over the intervals (-infinite;-2) and (-1;+infinite), and f(x) is concave
for x between (-2 , -1).