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

function is convex at

Given function:

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

To determine whether ( f(x) = 1 - xe^{-3x} ) is concave or convex at ( x = -2 ), we need to find the second derivative of ( f(x) ) and then evaluate it at ( x = -2 ). If the second derivative is positive at ( x = -2 ), the function is concave upward (convex), and if it's negative, the function is concave downward.

The first derivative of ( f(x) ) is: [ f'(x) = e^{-3x}(3x - 1) ]

The second derivative of ( f(x) ) is: [ f''(x) = 9xe^{-3x} - 6e^{-3x} ]

Substituting ( x = -2 ) into ( f''(x) ): [ f''(-2) = 9(-2)e^{-3(-2)} - 6e^{-3(-2)} ]

Solving this expression: [ f''(-2) = -18e^{6} - 6e^{6} ]

[ f''(-2) = -24e^{6} ]

Since ( -24e^{6} ) is negative, the function ( f(x) ) is concave downward at ( x = -2 ).

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

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.

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.

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.

- What are the points of inflection of #f(x)=x^2-x^(1/2) #?
- What are the points of inflection of #f(x)=3ln(x^(2) +2) -2x #?
- For what values of x is #f(x)=(4x)/(x^2-1)# concave or convex?
- How do you find the inflection points of the graph of the function: #f(x)=x^4-6x^3#?
- What are the points of inflection, if any, of #f(x)=3x^5 - 5x^4 #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7