# Is #f(x)=3/x-2x# concave or convex at #x=9/4#?

Convex

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To determine whether the function ( f(x) = \frac{3}{x} - 2x ) is concave or convex at ( x = \frac{9}{4} ), we need to find the second derivative and evaluate it at ( x = \frac{9}{4} ).

First, find the first derivative of ( f(x) ), then find the second derivative.

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

Now, find the second derivative:

( f''(x) = \frac{6}{x^3} )

At ( x = \frac{9}{4} ), ( f''\left(\frac{9}{4}\right) = \frac{6}{\left(\frac{9}{4}\right)^3} )

Calculate ( \frac{9}{4} ) cubed and then calculate ( \frac{6}{\left(\frac{9}{4}\right)^3} ).

If ( f''\left(\frac{9}{4}\right) > 0 ), the function is concave up at ( x = \frac{9}{4} ).

If ( f''\left(\frac{9}{4}\right) < 0 ), the function is concave down at ( x = \frac{9}{4} ).

If ( f''\left(\frac{9}{4}\right) = 0 ), the test is inconclusive, and we may need additional information to determine concavity.

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