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

Verified step by step guidance

1. 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.
2. 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}.
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.
4. 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}.
5. 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.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x) and output (f(x)). Understanding how to identify key features such as intercepts, symmetry, and asymptotes is essential for creating an accurate representation of the function.

Polynomial Functions

The function f(x) = 2 - 2x^(2/3) + x^(4/3) is a polynomial function, which is a sum of terms consisting of variables raised to non-negative integer powers. Recognizing the degree and leading coefficient of the polynomial helps in determining the end behavior and shape of the graph.

Critical Points and Derivatives

Finding critical points involves calculating the derivative of the function and setting it to zero to identify where the function's slope changes. Analyzing these points helps in determining local maxima, minima, and points of inflection, which are crucial for sketching the complete graph accurately.