Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y=x^4-8x^3+18x^2-16x+5$

Short answer

The function has its relative extrema points at x=[1, 2, 4] and points of inflection at x=[1,3].

Step-by-step solution

Derivation

Differentiate the function to find the first derivative. The derivative of $y=x^4-8x^3+18x^2-16x+5$ is $y^{\prime }=4x^3-24x^2+36x-16$. This helps to find points of relative extrema.

Find Critical Points

Set the first derivative equal to zero and solve for $x$ to find the critical points. These points are the potential locations of relative extrema. Solving the equation $4x^3-24x^2+36x-16=0$ for $x$ gives the solutions $x=1$, $x=2$, and $x=4$.

Second Derivation

Find the second derivative by differentiating the first derivative. The second derivative of $y^{\prime }=4x^3-24x^2+36x-16$ is $y^{″}=12x^2-48x+36$. This will help indicate points of inflection.

Identify Inflection Points

Set the second derivative equal to zero and solve for $x$ to identify potential points of inflection. Solving the equation $12x^2-48x+36=0$ for $x$ yields $x=1$ and $x=3$.

Check for Extrema and Inflection points

Substitute the critical points in the original function to find the actual maximum and minimum points. Also substitute inflection points to find the points of inflection. Choose a scale that displays these points properly on the graph.

Sketch the graph

Using the points found, sketch the graph of the function. Mark the points of relative extrema and points of inflection on the graph.