Understanding Concave Up Functions | The Role of Positive Second Derivatives in Calculus

If f(x) is concave up, then f”(x) is?

If a function f(x) is concave up, it means that its graph curves upwards, resembling a “U” shape

If a function f(x) is concave up, it means that its graph curves upwards, resembling a “U” shape. In terms of calculus, this implies that the second derivative of f(x), denoted as f”(x), is positive.

The second derivative of a function measures the rate at which the slope of the function is changing. If f”(x) is positive, it indicates that the slope of f(x) is increasing as x moves along the function. This increase in slope leads to the concave up shape of the graph.

To understand this concept more intuitively, consider the graph of a simple quadratic function, such as f(x) = x^2. Taking the derivative of f(x) gives f'(x) = 2x, and the second derivative f”(x) is always positive (2). As x increases, the slope of the graph (represented by f'(x)) continually increases, causing the graph to curve upwards and exhibit a concave up shape.

In summary, if a function f(x) is concave up, the second derivative f”(x) will be positive. This positive second derivative indicates that the slope of the function is increasing, leading to the upward curvature of the graph.

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