How to find whether a curve is concave or convex

The differential coefficient, or rate of change of tan a with respect to x for the concave curve APB continually decreases. Hence d(tan a)/dx is negative, or d(tan a) d2y 

If a function, y = f(x), increases with increasing values of x, dyjdx is positive while if the function, y = f(x), decreases with increasing values of x, dyjdx is negative. 

Along the convex part of the curve BQG, tan a regularly increases in value. Let us take numbers. Suppose a2 = 135°, ag = 45°, then tana2 = - 1 and tana3 = + 1. Hence as you pass along the curve from B to Q, tan a increases in value from - 1 to 0. At the point Q, tan a = 0, and from Q to C, tan a continually 

Hence a curve is concave or convex upwards, according as the second differential coefficient is positive or negative. 

I have assumed that the curve is on the positive side of the -axis when the curve lies on the negative side, assume the a -axis to be displaced parallel with itself until the above condition is attained. A more general rule, which evades the above limitation, is proved in the regular text-books. The proof is of little importance for our purpose. The rule is to the effect that a curve is concave or convex upwards according as the product of the ordinate of the curve and the second differential coefficient, i.e., according as yd y/dx2 is positive or negative . 

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