If f(x) is concave down, then f”(x) is?
If a function f(x) is concave down, it means that its graph is shaped like a downward-facing curve
If a function f(x) is concave down, it means that its graph is shaped like a downward-facing curve. In terms of the second derivative, f”(x), if f(x) is concave down, then f”(x) will be negative.
Mathematically, the concavity of a function can be determined by looking at the sign of its second derivative. If f”(x) is positive, it means that the function is concave up, with a graph that is shaped like a upward-facing curve. On the other hand, if f”(x) is negative, it indicates that the function is concave down, with a graph that is shaped like a downward-facing curve.
To further illustrate this, let’s consider an example. Suppose we have a function f(x) = -x^2. If we take the first derivative of this function, we get f'(x) = -2x. And if we take the second derivative, we get f”(x) = -2. Since f”(x) is negative (-2), it means that the function f(x) = -x^2 is concave down.
In summary, if a function f(x) is concave down, then its second derivative f”(x) will be negative.
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