f”(x) is positive
f'(x) is increasing and f(x) is concave up U
If the second derivative of a function, f”(x), is positive, it means that the function is concave up. In other words, its graph is bending upwards, like a smile.
The concavity of a function is an important concept in calculus, as it provides information about the relative rate of change of the function. When the second derivative of a function is positive, it indicates that the rate of change of the slope of the function is increasing.
For example, if we have a parabola y = x^2 and take the second derivative, we get y” = 2. Since this value is positive, we know that the graph of the parabola is concave up. In other words, the slope of the tangent line to the parabola is increasing as we move from left to right along the curve.
In practical terms, this means that the function is increasing at an accelerating rate. For example, if we are modeling a physical system, such as the motion of an object, a positive second derivative indicates that the object is accelerating at an increasing rate, implying that a greater force or energy is being applied to it.
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