Questions: Approximate the coordinates of each turning point by graphing (f(x)) in the standard viewing rectangle. [ f(x)=x^4-4 x^3+4 x^2-4 x+1 ] A. (2.32,-7.73) B. (0,1) C. (2.73,-5.95) D. (0.23,0.25) and (2.73,-5.95)

Solution Steps

To approximate the coordinates of the turning points of the function \( f(x) = x^4 - 4x^3 + 4x^2 - 4x + 1 \), we can graph the function and visually inspect the graph to identify the turning points. We will use Python's matplotlib library to plot the function and numpy to handle the calculations.

We are given the function: \[ f(x) = x^4 - 4x^3 + 4x^2 - 4x + 1 \]

To find the turning points, we need to find the first derivative of the function and set it to zero: \[ f'(x) = \frac{d}{dx}(x^4 - 4x^3 + 4x^2 - 4x + 1) \] \[ f'(x) = 4x^3 - 12x^2 + 8x - 4 \]

Set the first derivative equal to zero to find the critical points: \[ 4x^3 - 12x^2 + 8x - 4 = 0 \]

This is a cubic equation, and solving it analytically can be complex. We will use numerical methods or graphing tools to approximate the roots.

Using a graphing tool, plot the function \( f(x) = x^4 - 4x^3 + 4x^2 - 4x + 1 \) to identify the turning points.

From the graph, we can approximate the coordinates of the turning points. The graph shows turning points around the coordinates:

• \((0.23, 0.25)\)
• \((2.73, -5.95)\)

Final Answer