Answer to Question #101525 in Calculus for Ravi

Question

Find the point of inflexion of the curve.

Answer 1

Any point at which concavity changes is called an inflection point for the function.

For example, a parabola $f(x) = ax^2 + bx + c$ has no inflection points, because its graph is always concave up or concave down.

Steps to find the point of inflection:

1) Finding the Derivatives of a Function${\displaystyle f^{\prime }(x)} \ or \ {\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} x}}}$

2) Differentiate again. The second derivative is the derivative of the derivative, and is denoted as${\displaystyle f^{\prime \prime }(x)} \ or\ {\displaystyle {\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}}.}$

3) Set the second derivative equal to zero, and solve the resulting equation. Your answer will be a possible inflection point

4) Check if the second derivative changes sign at the candidate point. If the sign of the second derivative changes as you pass through the candidate inflection point, then there exists an inflection point. If the sign does not change, then there exists no inflection point