Derivatives And Critical Points: A Guide To Finding The Maximum And Minimum Of A Function.

Find derivative. Find critical points.When f ‘(x) is positive – increasing, negative – decreasing

Find the interval where f(x) is increasing/decreasing given an equation.

To find the derivative of a function, we need to use differentiation rules. Let’s say we have a function f(x), then the derivative of the function with respect to x is denoted by f ‘(x).

To find the critical points of a function, we set the derivative to zero and solve for x. These critical points tell us where the function may have a maximum or minimum.

To determine whether a function is increasing or decreasing, we look at the sign of the derivative. If f ‘(x) is positive, then the function is increasing. If f ‘(x) is negative, then the function is decreasing.

Here is an example to demonstrate these concepts:

Let’s say we have the function f(x) = x^3 – 3x^2 + 2x.

To find the derivative of this function, we need to use the power rule of differentiation. The power rule states that if we have a function of the form x^n, then its derivative is given by nx^(n-1). Therefore,

f ‘(x) = 3x^2 – 6x + 2.

To find the critical points of the function, we need to set the derivative to zero and solve for x.

3x^2 – 6x + 2 = 0.

Using the quadratic formula, we find that the solutions are:

x = (6 ± sqrt(36 – 24))/6

= (6 ± sqrt(12))/6

= 1 ± sqrt(3)/3

Therefore, the critical points of the function f(x) are x = 1 + sqrt(3)/3 and x = 1 – sqrt(3)/3.

To determine whether the function is increasing or decreasing, we need to look at the sign of the derivative.

When x < 1 - sqrt(3)/3, f '(x) is negative, so the function is decreasing. When 1 - sqrt(3)/3 < x < 1 + sqrt(3)/3, f '(x) is positive, so the function is increasing. When x > 1 + sqrt(3)/3, f ‘(x) is negative, so the function is decreasing.

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