# Is #f(x)=x^3-2x+7 # concave or convex at #x=2 #?

convex at x = 2

To determine if a function f(x) is concave/convex at x = a we require to find the value of f''(a).

• If f''(a) > 0 then f(x) is convex at x = a

• If f''(a) < 0 then f(x) is concave at x = a

Since f''(2) > 0 then f(x) is convex at x = 2. graph{x^3-2x+7 [-22.5, 22.5, -11.25, 11.25]}

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To determine the concavity of the function ( f(x) = x^3 - 2x + 7 ) at ( x = 2 ), we need to find the second derivative of the function and then evaluate it at ( x = 2 ).

First, find the first derivative of ( f(x) ): [ f'(x) = 3x^2 - 2 ]

Now, find the second derivative of ( f(x) ): [ f''(x) = 6x ]

Evaluate the second derivative at ( x = 2 ): [ f''(2) = 6(2) = 12 ]

Since the second derivative is positive at ( x = 2 ), the function is concave up at that point.

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