Question

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=x+\frac{32}{x^{2}}\)

Short answer

The x-intercept is (-4,0). The relative extrema is at point (4,5). No points of inflection exist. The vertical asymptote is \(x=0\) and the horizontal asymptote is \(y=x\). The domain of the function is all real numbers except 0.

Step-by-step solution

Find the Intercepts

To get the x-intercept, set \(y=0\) and solve for \(x\), and to find the y-intercept, set \(x=0\) and solve for \(y\). Unfortunately, setting \(x=0\) yields an undefined result because we cannot divide by zero. So there's no y-intercept. For \(x\), find the solution to this equation \(0=x+\frac{32}{x^{2}}\). This simplifies to \(x^{3}=-32\), so \(x=-4\). Thus the x-intercept is (-4,0).

Calculate the Relative Extrema

Relative extrema are the local minimum and maximum points of a function. To find these, you need to calculate the derivative, set it equal to zero, and solve for \(x\). The derived function of \(y=x+\frac{32}{x^{2}}\) is \(y'=1-\frac{64}{x^{3}}\). Solving \(y'=0\) for \(x\), yields the cube root of 64, which simplifies to 4. Substitute \(x=4\) back into the original function to find \(y=5\). Thus, the relative extrema point is (4,5).

Identify the Points of Inflection

Points of inflection are found by deriving the function twice, setting the second derivative to zero, and solving for \(x\). The outcome is \(y''=\frac{192}{x^{4}}\). As the second derivative never equals 0, there are no points of inflection for this function.

Determine the Asymptotes

Vertical asymptotes occur when the function tends towards infinity, which is when \(x=0\). Horizontal asymptotes are the values the function approaches as \(x\) becomes infinitely large or small. Since as \(x\) approaches infinity or negative infinity, the function tends towards \(x\), hence the line \(y=x\) is the horizontal asymptote.

State the Domain of the Function

The domain of a function is the set of all permissible \(x\) values. In this equation, \(x\) can be any real number except zero (because we can't divide by zero) so the domain is \(x ≠ 0\), or (-∞,0) U (0, ∞).