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Chapter 9: Problem 39

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=x \sqrt{4-x}\)

Short Answer

Expert verified
The x and y intercepts are at (0,0) and (4,0). There is a relative maximum at \(x = 4/3\). There are no points of inflection and asymptotes. The domain of the function is \([0, 4]\).

Step by step solution

01

Find the intercepts

The intercepts are the points where the graph of the function intersects the axes. For the x-intercept, set \(y = 0\) and solve for \(x\). Hence, the x-intercepts are (0,0) and (4,0). For the y-intercept, set \(x = 0\) and solve for \(y\). Hence, the y-intercept is (0,0).
02

Find the extrema

Extrema occur where the derivative is zero or undefined. First, find the derivative \(y'\). Derivative of \(y=x \sqrt{4-x}\) is \(y' = \sqrt{4-x} - \frac{x}{2 \sqrt{4-x}}\) using the product rule and chain rule. Set the derivative equal to zero to find critical points which gives \(x = 4/3\). Then use the second derivative test to find out if it's a relative minimum or maximum. The second derivative \(y'' = -\frac{3}{4 \sqrt{(4-x)^3}}\), plug \(x = 4/3\) to check, which yields a negative result, indicating a relative maximum.
03

Find points of inflection

Inflection points occur where the second derivative changes sign. In our case, it's not possible for \(y''\) to change signs since it's always negative when defined, so there are no points of inflection.
04

Determine asymptotes

Vertical asymptotes occur when the function goes to positive or negative infinity as \(x\) approaches a certain value, which doesn鈥檛 happen for this function. Horizontal asymptotes occur when \(y\) approaches a certain value as \(x\) goes to positive or negative infinity, but the function goes to \(0\) as \(x\) approaches positive infinity, and is undefined for negative \(x\), with no horizontal asymptotes in existence.
05

Find the domain

The domain of the function is the set of x-values for which the function is defined. Since the function includes a square root, the quantity inside the square root must be greater than or equal to zero for the function to be real. Hence, the domain is \(0 \leq x \leq 4\).

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Most popular questions from this chapter

A state game commission introduces 50 deer into newly acquired state game lands. The population \(N\) of the herd can be modeled by \(N=\frac{10(5+3 t)}{1+0.04 t}\) where \(t\) is the time in years. Use differentials to approximate the change in the herd size from \(t=5\) to \(t=6\).

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x}{\sqrt{x^{2}-4}}\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=3 x^{4}+4 x^{3}\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{3}+3 x^{2}+9 x-2\)

Find the differential \(d y\). \(y=3 x^{2}-4\)
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