Write down the equation of the curve, in terms of Y and X.
For example, consider the curve Y = f(X) = X^2 and the curve Y = g(X) = sin(X)
Calculate the second derivatives for the functions.
From our example:
f(X) = X^2
f'(X) = 2x [first derivative]
f''(X) = 2 [second derivative]
g(X) = sin(X)
g'(X) = cos(X) [first derivative]
g''(X) = -sin(X) [second derivative]
Check the value of the second derivative on the interval you wish to evaluate. Concavity is found on parts of a curve so an interval is required to determine concavity. If the second derivative is greater than zero (positive), the curve is concave up. If the second derivative is less than zero (negative), the curve is concave down.
From the example:
f''(X) = 2; since the second derivative is 2 regardless of the value of X, the curve is concave up ( 2 > 0 ).
g''(X) = -sin(X) on the interval ] 0, (PI/2) [ and the interval ] (PI/2), PI [
Since -sin(X) > 0 on the interval ] 0, (PI/2) [; therefore the curve is concave up.
Since -sin(X) < 0 on the interval ] (PI/2), PI [; thus the curve is concave down.
Check the value of the second derivative for zeroes. If the second derivative has a value equal to zero, then the curve has an inflection point.
From the example:
f''(X) = 2; since the second derivative is 2 regardless of the value of X, there are no inflection points on this curve.
g''(X) = -sin(X) on the point X=0.
Since g''(0) = -sin(0) = 0, there is an inflection point at X = 0.