Mastering Calculus: Identifying Local Maximum At X Using Derivatives

When f ‘(x) changes from positive to negative, f(x) has a

relative maximum

local maximum at x.

Explanation:

The derivative of a function f(x) represents the rate of change of the function f(x) with respect to x. If the value of the derivative f ‘(x) is positive at x, then the function f(x) is increasing at x, i.e., the slope of the tangent at x is positive. Similarly, if f ‘(x) is negative at x, then the function f(x) is decreasing at x, i.e., the slope of the tangent at x is negative.

Now, if f ‘(x) changes from positive to negative at x, it means that the function f(x) is changing from increasing to decreasing at x. This implies that the slope of the tangent at x changes from positive to negative, which means that the tangent line changes from an uphill slope to a downhill slope at x. At the point where the slope of the tangent changes sign, the function f(x) has either a local maximum or a local minimum, depending on whether the change is from positive to negative or negative to positive, respectively.

In this case, since the derivative f ‘(x) changes from positive to negative, it implies that the function f(x) changes from increasing to decreasing, and therefore, f(x) has a local maximum at x. This is because the maximum point represents a point where the function changes from increasing to decreasing, and at this point, the tangent slope changes from positive to negative.

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