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For what values of x is #f(x)=1/(x-3)# concave or convex?

Answer 1

Charles Anctil

Concave UP on the interval (#-oo, 3#)
Concave DOWN on the interval (#3, oo#)

Take the first derivative of f(x)

#f'(x) = -1/(x-3)^2#

Then second derivative:

#f''(x) = 2/(x-3)^3#

Find the points where #f''(x) = 0# or is undefined.

#f(x)# is undefined when #x=3#

Plug in a number #x>3# and you will see the function will be positive; thus the function is concave up.

Since the multiplicity of the function #(x-3)^3# is an odd function, the values for #x<3# will be negative; thus the function will be concave down.

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Answer 2

Charles Arocha

To determine where the function ( f(x) = \frac{1}{x-3} ) is concave or convex, we need to analyze its second derivative.

The second derivative of ( f(x) ) is:

[ f''(x) = \frac{2}{(x-3)^3} ]

For a function to be concave up (convex) on an interval, its second derivative must be positive on that interval. For ( f(x) ), ( f''(x) ) is positive for all ( x ) except ( x = 3 ) where it is undefined.

Therefore, ( f(x) ) is concave up (convex) for all ( x ) except at ( x = 3 ).

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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.