Understanding Concavity: Exploring the Relationship between the Second Derivative and Graph Shape

If f(x) is concave up, then f”(x) is?

If a function f(x) is concave up, it means that its graph curves upward, forming a “smile” shape

If a function f(x) is concave up, it means that its graph curves upward, forming a “smile” shape. In terms of the second derivative f”(x), this means that f”(x) is positive.

To understand why, let’s recall the relationship between the concavity of a function and its second derivative.

The concavity of a function is determined by the sign of its second derivative. If f”(x) is positive, it indicates that the graph of f(x) is concave up. Conversely, if f”(x) is negative, it indicates that the graph of f(x) is concave down.

Mathematically, this can be further explained:

– If f”(x) > 0, then the function f(x) is concave up.
– If f”(x) < 0, then the function f(x) is concave down. For example, let's consider the function f(x) = x^2. Its first derivative is f'(x) = 2x, and the second derivative is f''(x) = 2. Since f''(x) = 2 > 0 for all x, it means that the function f(x) = x^2 is concave up.

So, in general, if a function f(x) is concave up, then its second derivative f”(x) will be positive.

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