# Is #f(x) = -3x^4+x^3 - 4x^2+x-4# concave or convex at #x=11#?

The answer lies in the sign of the second derivative: the function is convex if the second derivative is positive, and concave otherwise. So, let's compute it.

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To determine whether the function (f(x) = -3x^4 + x^3 - 4x^2 + x - 4) is concave or convex at (x = 11), we need to examine the second derivative of the function at that point.

First, we find the first and second derivatives of the function (f(x)): [f'(x) = -12x^3 + 3x^2 - 8x + 1] [f''(x) = -36x^2 + 6x - 8]

Now, we evaluate (f''(11)) to determine the concavity/convexity at (x = 11): [f''(11) = -36(11)^2 + 6(11) - 8]

After computing this value, we can conclude:

- If (f''(11) > 0), then (f(x)) is convex at (x = 11).
- If (f''(11) < 0), then (f(x)) is concave at (x = 11).
- If (f''(11) = 0), the test is inconclusive, and we may need to apply higher-order tests or examine the behavior around the point more closely.

Substitute the value of (x = 11) into (f''(x)), calculate the result, and determine whether it's positive or negative to ascertain whether (f(x)) is concave or convex at (x = 11).

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