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

and

graph{x(x-2)(x+3)(x-1) [-4, 3, -26, 10]}

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To determine the concavity or convexity of the function ( f(x) = x(x-2)(x+3)(x-1) ) at ( x = 1 ), we need to evaluate the second derivative of the function at ( x = 1 ). If the second derivative is positive at ( x = 1 ), then the function is concave up (convex) at that point. If the second derivative is negative at ( x = 1 ), then the function is concave down at that point.

The second derivative of ( f(x) ) with respect to ( x ) is given by:

[ f''(x) = 12x^2 - 6x - 6 ]

Now, plug in ( x = 1 ):

[ f''(1) = 12(1)^2 - 6(1) - 6 = 12 - 6 - 6 = 0 ]

Since the second derivative at ( x = 1 ) is zero, we cannot determine the concavity or convexity of the function at that point using the second derivative test. We need to employ other methods such as the first derivative test or analyzing the behavior of the function around ( x = 1 ).

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