Understanding the Shape of a Curve | The Implications of a Negative Second Derivative (y” < 0)

When y” < 0, the curve is

When y” < 0, it means that the second derivative of the function y(x) is negative

When y” < 0, it means that the second derivative of the function y(x) is negative. In other words, the rate of change of the slope of the curve is negative. This has important implications for the shape of the curve. If we consider the graph of y(x), a curve with a negative second derivative implies that the curve is concave down. This means that the curve is shaped like a frown or a smile that is upside down. Mathematically, if y'' < 0, it indicates that the average slope of the curve is decreasing as x increases. Graphically, this can be visualized as the curve initially becoming steeper, but then gradually leveling off. The concave down shape can be observed when the curve is bending downward. To better understand this, let's consider an example. Suppose we have a quadratic function y(x) = x^2. The first derivative of this function is y'(x) = 2x, and the second derivative is y''(x) = 2. Since y''(x) = 2 > 0, this means the curve is concave up. On the other hand, if we consider the function y(x) = -x^2, the second derivative is y”(x) = -2.

Since y”(x) = -2 < 0, this means the curve is concave down. In this case, the function represents a symmetric downward opening parabola. In summary, when y'' < 0, the curve is concave down, and the slope of the curve is decreasing as x increases. This information provides insights into the shape and behavior of the curve.

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