# Find the Concavity f(x)=e^(2x)+e^(-x)

Find the inflection points.
Find the second derivative.
Find the first derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Move to the left of .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Move to the left of .
Rewrite as .
Find the second derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Move to the left of .
Multiply by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Move to the left of .
Rewrite as .
Multiply by .
Multiply by .
The second derivative of with respect to is .
Graph each side of the equation. The solution is the x-value of the point of intersection.
No solution
No values found that can make the second derivative equal to .
No Inflection Points
No Inflection Points
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
The graph is concave up because the second derivative is positive.
The graph is concave up
Find the Concavity f(x)=e^(2x)+e^(-x)