f'(x) changes from positive to negative
f(x) has a relative maximum
When f'(x) changes from positive to negative, it means that the slope of the function is changing from increasing to decreasing as x increases. In other words, the function is going from increasing to decreasing.
This point where the slope changes from positive to negative is known as a local maximum. At this point, the slope of the tangent line to the function is zero. The graph of the function looks like a hill or a peak at this point.
For example, consider the function f(x) = x^3 – 3x^2 + 2x. The derivative of this function is f'(x) = 3x^2 – 6x + 2. When we set f'(x) equal to zero and solve for x, we get x = 1 ± √(3)/3. These are the two points where the slope changes from positive to negative.
At x = 1 – √(3)/3, the function has a local maximum. At this point, the slope of the tangent line is zero and the graph of the function looks like a peak. As x increases beyond this point, the slope of the tangent line becomes negative and the function starts to decrease. Similarly, at x = 1 + √(3)/3, the function has a local minimum. At this point, the slope of the tangent line is zero and the graph of the function looks like a valley. As x increases beyond this point, the slope of the tangent line becomes positive and the function starts to increase.
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