Home > Calculus > Analyzing Concavity of a Function

Is #f(x)=1-e^(3x)# concave or convex at #x=2#?

Answer 1

Scarlett Allison

Find the second derivative at #x=2#

If the second derivative is positive , it's concave up . Likewise, if it is negative , it will be concave down .

#f'=-3e^(3x)#

#f''=-9e^(3x)#

#f''(2) <0#, so it is concave down or convex

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Evelyn Agard

To determine whether ( f(x) = 1 - e^{3x} ) is concave or convex at ( x = 2 ), we need to analyze the second derivative of the function at that point.

Find the second derivative of the function: [ f(x) = 1 - e^{3x} ] [ f'(x) = -3e^{3x} ] [ f''(x) = -9e^{3x} ]
Evaluate the second derivative at ( x = 2 ): [ f''(2) = -9e^{3(2)} = -9e^6 ]

Since the second derivative is negative at ( x = 2 ), the function is concave downwards at that point.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 3

Christopher Agresta

To determine whether the function ( f(x) = 1 - e^{3x} ) is concave or convex at ( x = 2 ), we need to analyze the second derivative of the function at that point.

First, find the second derivative of the function: [ f(x) = 1 - e^{3x} ] [ f'(x) = -3e^{3x} ] [ f''(x) = -9e^{3x} ]

Now evaluate the second derivative at ( x = 2 ): [ f''(2) = -9e^{3(2)} = -9e^6 ]

Since the second derivative is negative at ( x = 2 ), ( f(x) ) is concave downward at ( x = 2 ).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.