Use the guidelines of this section to make a complete graph of f.
f(x) = 2 - 2x^2/3 + x^4/3

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Identify the domain of the function f(x) = 2 - 2x^{2/3} + x^{4/3}. Since the function involves fractional exponents, check for any restrictions. In this case, the domain is all real numbers because the exponents are positive and the base x can be any real number.

Find the first derivative f'(x) to determine the critical points and analyze the increasing or decreasing behavior of the function. Use the power rule for derivatives: f'(x) = -\(\frac{4}{3}\)x^{-1/3} + \(\frac{4}{3}\)x^{1/3}.

Set the first derivative f'(x) equal to zero to find critical points: -\(\frac{4}{3}\)x^{-1/3} + \(\frac{4}{3}\)x^{1/3} = 0. Solve this equation to find the values of x where the slope of the tangent is zero.

Find the second derivative f''(x) to determine the concavity and points of inflection. Differentiate f'(x) to get f''(x) = \(\frac{4}{9}\)x^{-4/3} - \(\frac{4}{9}\)x^{-2/3}.

Analyze the behavior of f(x) as x approaches positive and negative infinity to understand the end behavior of the graph. Also, use the second derivative test to confirm the nature of the critical points found in step 3, and identify any points of inflection from the second derivative.

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