How to find points of inflexion

From the above principles it is clearly necessary, in order to locate a point of inflexion, to find a value of x, for which tana assumes a maximum or a minimum value. But 

Hence the rule In order to find a point of inflexion we must equate the second differential coefficient of the function to zero find the value of x which satisfies these conditions and test if the t second differential coefficient does really change sign by substituting in the second differential coefficient a value of a a little greater and one a little less than the critical value. If there is no change of sign we are not dealing with a point of inflexion 

Examples.—(1) Show that the curve y=a+ x-by has a point of inflexion at the point y = a, x — b. Differentiating twice we get d yldx2 = 6(x - b). Equating this to zero we get x — b by substituting x=b in the original equation, we get y= a. When x = b - 1 the second differential coefficient is negative, when x=b+l the second differential coefficient is positive. Hence there is an inflexion at the point (b, a). See Fig. 70, page 158. 

---