Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=2 x^{4}-3 x^{3}+5 x^{2}-20 x+40 $$

Short answer

To plot the graph of the function \(f(x) = 2x^4 - 3x^3 + 5x^2 - 20x + 40\), first identify the intercepts (\(x\)-intercepts must be approximated numerically), and then find the approximate locations of critical points by solving \(f'(x)=0\) and inflection points by solving \(f''(x)=0\). Finally, choose an appropriate viewing window to capture the identified points and plot the graph using a graphing calculator or software like Desmos.

Step-by-step solution

Identify the domain and range

As \(f(x)\) is a polynomial function, its domain is all real numbers i.e., \(-\infty < x < \infty\). Also, since it is a quartic function with a positive leading coefficient, the function's range will be from some minimum value to infinity i.e., \([y_{min}, \infty)\).

Find x and y-intercepts

To find the x-intercepts, we set the function \(f(x)\) equal to 0: \(2x^4 - 3x^3 + 5x^2 - 20x + 40 = 0\). In this case, finding the roots exactly might be challenging as it is a quartic equation. We can approximate the roots using a graphing calculator or a software like Desmos. For the y-intercept, we set \(x=0\) in the function \(f(x)\): \(f(0) = 2(0)^4 - 3(0)^3 + 5(0)^2 - 20(0) + 40\), which simplifies to \(f(0) = 40\), so the y-intercept is (0, 40).

Find critical and inflection points

To find critical points, we calculate the first derivative \(f'(x)\) and set it equal to 0: \(f'(x) = 8x^3 - 9x^2 + 10x - 20\). Solving the cubic equation \(8x^3 - 9x^2 + 10x - 20 = 0\), may be difficult, so we can use a numeric solver to find the approximate locations of the critical points. For inflection points, we take the second derivative \(f''(x)\) and set it equal to 0: \(f''(x) = 24x^2 - 18x + 10\). Solving the quadratic equation \(24x^2 - 18x + 10 = 0\), may not have exact solutions, so we can also use a numeric solver to find the approximate locations of the inflection points.

Prepare for graphing

With the x and y-intercepts, critical points, and inflection points found in previous steps, we can now use a graphing calculator, a software like Desmos, or hand-sketch the graph in an appropriate viewing window. The viewing window will depend on the critical points and inflection points' location, so choose one that encapsulates those as well as the x and y-intercepts. To plot the graph of the function \(f(x) = 2x^4 - 3x^3 + 5x^2 - 20x + 40\), we have roughly identified the important elements for sketching the graph, such as domain, range, intercepts, critical points, and inflection points. Now, you can plot the graph of the function using a graphing calculator or software like Desmos, where you can fine-tune the exact critical and inflection points, as required.