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For what values of x is #f(x)= x + 1/x # concave or convex?

Answer 1

Christopher Agresta

Concave or convex according as x < or > 0.

#(d^2/dx^2)f( x )= 2/x^3 < 0 or > 0 according as x < 0 or > 0..

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Answer 2

Ezra Adams

To determine where the function ( f(x) = x + \frac{1}{x} ) is concave or convex, we need to find its second derivative and examine its sign.

First, find the first derivative: [ f'(x) = 1 - \frac{1}{x^2} ]

Then, find the second derivative: [ f''(x) = \frac{2}{x^3} ]

Now, analyze the sign of the second derivative:

( f''(x) > 0 ) when ( x > 0 ). In this interval, the function is convex.
( f''(x) < 0 ) when ( x < 0 ). In this interval, the function is concave.

So, the function ( f(x) = x + \frac{1}{x} ) is convex for ( x > 0 ) and concave for ( x < 0 ).

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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.