# Is #f(x)=4x^5+2x^3-2x^2+2x+8# concave or convex at #x=-3#?

concave at x = -3

To determine if a function is concave/convex at f(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''(-3) < 0 , then f(x) is concave at x = -3

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To determine if the function ( f(x) = 4x^5 + 2x^3 - 2x^2 + 2x + 8 ) is concave or convex at ( x = -3 ), we need to evaluate the second derivative of the function at that point. If the second derivative is positive, the function is convex; if it is negative, the function is concave.

( f''(x) = 60x^3 + 6x - 4 )

( f''(-3) = 60(-3)^3 + 6(-3) - 4 = -540 - 18 - 4 = -562 )

Since the second derivative is negative ( (-562) ), the function is concave at ( x = -3 ).

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