To find inflection points and concavity, take the second derivative and set equal to 0. Solve for x, then test a value on the left of the value for x and one on the right. If the second derivative of this value is less than zero, it's concave down on that interval, if the second derivative of this value is greater than zero, then it's concave up.

For example:

f(x) = 3x^3 + 2x^2 + 5x + 6

first derivative :   f ' (x) = 9x^2 + 4x + 5

second derivative :  f " (x) = 18x + 4

set the second derivative equal to 0 and solve for x

18x + 4 = 0

18x = -4

x = -4/18 = -2/9

Test a value to the left of -2/9,  I choose -1

f " (-1) = -14

Test value to the right of -2/9, I choose 0

f " (0) = 4

Since f " (-1) is negative, the concavity is downward from negative infinity to -2/9

Since f " (0) is positive, the concavity is upward from -2/9 to infinity.