# Is #f(x)=x-sqrt(x^3-3x)# increasing or decreasing at #x=2#?

Decreasing.

It can be verified by drawing graph of the function using the graphing tool. graph{y=x-sqrt(x^3-3x) [-3.437, 6.56, -2.76, 2.24]}

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To determine if ( f(x) = x - \sqrt{x^3 - 3x} ) is increasing or decreasing at ( x = 2 ), we need to evaluate the derivative of ( f(x) ) at that point.

The derivative of ( f(x) ) is ( f'(x) = 1 - \frac{3x^2 - 3}{2\sqrt{x^3 - 3x}} ).

Substituting ( x = 2 ) into ( f'(x) ), we get ( f'(2) = 1 - \frac{3(2)^2 - 3}{2\sqrt{(2)^3 - 3(2)}} ).

Evaluating this expression, we find ( f'(2) = 1 - \frac{9}{2\sqrt{2}} ).

Since ( \frac{9}{2\sqrt{2}} > 1 ), ( f'(2) < 0 ).

Therefore, ( f(x) ) is decreasing at ( x = 2 ).

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