How to find the concavity of trigger functions

Curvature behavior

Example of a function that is left and right curved

\ (f (x) = x ^ 3 - x ^ 2 \)

\ (f '(x) = 3x ^ 2 - 2x \)

\ (f '' (x) = 6x - 2 \)

If there is an \ (x \) in the 2nd derivative of the function, it is usually a function that has left-curved and right-curved areas. These ranges or intervals can be calculated by considering where the 2nd derivative is smaller (larger) zero.

When is the 2nd derivative less than zero?

\ [\ text {Approach:} 6x - 2 <0 \]

We now have to solve the above inequality for \ (x \).

\ [6x - 2 <0 \ quad | +2 \]

\ [6x <2 \ quad |: 6 \]

\ [x <\ frac {2} {6} \]

\ [x <\ frac {1} {3} \]

It follows:

\ [\ text {For} \ quad x <\ frac {1} {3} \ quad \ text {the function is right-curved.} \]

When is the 2nd derivative greater than zero?

\ [\ text {Approach:} 6x - 2> 0 \]

We now have to solve the above inequality for \ (x \).

\ [6x - 2> 0 \ quad | +2 \]

\ [6x> 2 \ quad |: 6 \]

\ [x> \ frac {2} {6} \]

\ [x> \ frac {1} {3} \]

It follows:

\ [\ text {For} \ quad x> \ frac {1} {3} \ quad \ text {the function is left-curved.} \]

The function \ (f (x) = x ^ 3-x ^ 2 \) is
for \ (x <\ frac {1} {3} \) curved to the right (concave) and
for \ (x> \ frac {1} {3} \) curved to the left (convex).

To illustrate the transition from concave to convex, a dashed line was drawn in at \ (x = \ frac {1} {3} \).

In this article we learned how to use the 2nd derivative to determine the curvature behavior of a function. The curvature behavior provides information about the areas in which a function is curved to the left (convex) or curved to the right (concave).