Home > Calculus > Analyzing Concavity of a Function

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

Answer 1

Convex

Considering locally convex a curve when it has second derivative negative valued, #(d^2f(x))/(dx^2)|_(x=-2) = -0.129572#

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Answer 2

Cameron Alexander

To determine the concavity of the function ( f(x) = \frac{x}{1 - x^3 e^{\frac{x}{3}}} ) at ( x = -2 ), we need to examine the second derivative. If the second derivative is positive, the function is concave up; if it's negative, the function is concave down.

The second derivative of ( f(x) ) can be calculated by taking the derivative of the first derivative:

[ f''(x) = \frac{d^2}{dx^2} \left(\frac{x}{1 - x^3 e^{\frac{x}{3}}}\right) ]

After finding the second derivative, we evaluate it at ( x = -2 ) to determine the concavity at that point.

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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.