Critical Points In Math: Explained With Examples

f'(x) changes from positive to negative

f(x) has a relative maximum

When f'(x) changes from positive to negative, it indicates that the slope or the rate of change of the function is decreasing. This means that the function is increasing at a slower rate as we move from left to right. At the point where f'(x) changes from positive to negative, the function reaches a relative maximum or a local peak.

To understand this better, let’s consider an example. Suppose we have a function f(x) = x^2 – 3x + 2. The derivative of this function is f'(x) = 2x – 3.

When f'(x) changes from positive to negative, it occurs at x = 3/2. Before x = 3/2, f'(x) is positive which indicates that the function is increasing. After x = 3/2, f'(x) is negative which indicates that the function is decreasing. This point, x = 3/2, corresponds to a relative maximum or a local peak of the function.

We can also visualize this using a graph. The graph of the function f(x) is a parabola that opens upwards. The point (3/2, 1/4) corresponds to the vertex of the parabola, which is a relative maximum. The slope of the tangent line at this point is zero, which is the minimum value of f'(x) in the neighborhood of x = 3/2. As we move away from this point in either direction, the derivative changes sign which means the function is changing from increasing to decreasing.

In summary, when f'(x) changes from positive to negative, it means that the function is reaching a relative maximum or a local peak and the slope or the rate of change of the function is decreasing.

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