# Is #f(x)=x^4-4x^3+x-4# concave or convex at #x=-1#?

convex

To find that out, we need to get the second derivative first.

Getting the first derivative.

We can easily find this using power rule.

Getting the second derivative.

Use power rule again.

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To determine whether ( f(x) = x^4 - 4x^3 + x - 4 ) is concave or convex at ( x = -1 ), we need to analyze the second derivative of the function at that point.

The second derivative of ( f(x) ) is denoted as ( f''(x) ), and it provides information about the concavity of the function.

First, we find the first derivative of ( f(x) ): [ f'(x) = 4x^3 - 12x^2 + 1 ]

Now, we find the second derivative: [ f''(x) = 12x^2 - 24x ]

To determine the concavity at ( x = -1 ), we plug ( x = -1 ) into ( f''(x) ): [ f''(-1) = 12(-1)^2 - 24(-1) = 12 - (-24) = 12 + 24 = 36 ]

Since ( f''(-1) = 36 ) and ( 36 > 0 ), the function is concave upward at ( x = -1 ).

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