Solved Examples on Concavity
Example 1: Determine the intervals where the function f(x)=x3−6x2+9x+15 is concave up and concave down.
First Derivative: f'(x)=3x2 -12x + 9
Second Derivative: f”(x)=6x-12
Find Critical Points for Concavity:
Set the second derivative equal to zero to find potential points of inflection:
6x-12=0
x=12/6=2
Test the Intervals Around the Critical Point:
for x<2 (eg. x=1):
f”(1)=6(1)-12=-6 (negative, concave down)
for x>2(eg. x=3):
f”(3)=6(3)-12=6 (positive, concave up)
Conclusion:
- The function is concave down on the interval (−∞,2).
- The function is concave up on the interval (2,∞).
Question -2: Determine the intervals where the function g(x)=2x4-4x3-24x2+48x is concave up and concave down.
First Derivative: g'(x)=8x3-12x2-48x+48
Second Derivative: g”(x)=24x2-24x-48
Find Critical Points for Concavity:
Set the second derivative equal to zero to find potential points of inflection:
24x2-24x-48=0
x2-x-2=0
(x-2)(x-1)=0
x=2,1
Test the Intervals Around the Critical Points:
For x<-1(eg. x=-2):
g”(2)=24(-2)2-24(-2)-48=96+48-48=96 (positive, concave up)
For -1<x<2 (eg. x=0):
g”(0)=24(0)2-24(0)-48=-48 (negative, concave down)
For x>2 (eg., x=3):
g”(3)=24(3)2-24(3)-48=216-72-48=96 (positive, concave up)
Conclusion:
The function is concave up on the intervals (−∞,−1) and (2,∞).
The function is concave down on the interval (−1,2).
Concavity and Points of Inflection
Concavity and points of inflection are the key concepts and basic fundamentals of calculus and mathematical analysis. It provides an insight into how curves behave and the shape of the functions. Where concavity helps us to understand the curving of a function, determining whether it is concave upward or downward, the point of inflection determines the point where the concavity changes, i.e., where either curve transforms from concave upward to concave downward or concave to convex, and vice versa. These concepts are essential in various mathematical applications, including curve sketching, optimization problems , and the study of differential equations.
In this article, we’ll shed lights on the definitions, properties, and practical implications of concavity and points of inflection.
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