Question

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

Short answer

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

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).

Find the extrema

Extrema occur where the derivative is zero or undefined. First, find the derivative $y^{\prime }$. Derivative of $y=x\sqrt{4-x}$ is $y^{\prime }=\sqrt{4-x}-\frac{x}{2\sqrt{4-x}}$ 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.

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.

Determine asymptotes

Vertical asymptotes occur when the function goes to positive or negative infinity as $x$ approaches a certain value, which doesn’t 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.

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 $0\le x\le 4$.