f(x) is concave up on an interval, if f”(x) is
positive on that interval
positive on that interval.
To understand why, let’s first clarify what it means for a function to be concave up. A function is said to be concave up on an interval if it curves upwards, resembling a “U” shape. On the other hand, a function is concave down on an interval if it curves downwards, resembling an upside-down “U” shape.
The second derivative of a function, denoted as f”(x), gives us information about the curvature of the function. If f”(x) is positive on an interval, it means that the slope of the function is increasing on that interval. Considering that the slope is the rate of change of the function, our function is getting steeper and steeper as x increases within that interval.
In terms of the graph, this translates to the graph of the function curving upwards on the interval. The function starts below the tangent line initially, and as x increases, it gradually approaches or surpasses the tangent line. This is why we call it concave up, as it resembles the shape of a cup facing upwards.
So, to summarize, if f”(x) is positive on an interval, it means that the function f(x) is concave up on that interval.
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