Find the extreme values of ƒ(x) = x³ - 3x², and find the area of the region enclosed by the graph of ƒ and the x-axis.

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To find the extreme values of the function \(f(x) = x^{3} - 3x^{2}\), first compute its derivative: \(f'(x) = 3x^{2} - 6x\).

Set the derivative equal to zero to find critical points: \(3x^{2} - 6x = 0\). Factor this equation to solve for \(x\).

Solve the factored equation to find the critical points, then use the second derivative test or analyze the sign changes of \(f'(x)\) around these points to determine whether each is a local maximum or minimum.

To find the area enclosed by the graph of \(f\) and the x-axis, first find the points where \(f(x) = 0\) by solving \(x^{3} - 3x^{2} = 0\).

Set up definite integrals of \(|f(x)|\) between the roots found, splitting the integral where the function crosses the x-axis if necessary, and evaluate these integrals to find the total enclosed area.

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Finding Extreme Values Using Derivatives

Extreme values of a function occur at critical points where the first derivative is zero or undefined. By differentiating ƒ(x) and solving ƒ'(x) = 0, we identify potential maxima or minima. The second derivative test or analyzing the sign changes of ƒ'(x) helps classify these points.

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Finding Global Extrema (Extreme Value Theorem)

Understanding the Graph and Roots of the Function

To find the area enclosed by the graph and the x-axis, it is essential to determine where the function crosses the x-axis (roots). These roots define the interval(s) over which the function lies above or below the x-axis, guiding the setup of definite integrals for area calculation.

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Calculating Area Using Definite Integrals

The area between the curve and the x-axis is found by integrating the absolute value of the function over the interval between its roots. This involves computing definite integrals and summing areas where the function is positive and negative, ensuring the total enclosed area is positive.

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