Sketch the graph and show all local extrema and inflection points.
-$= x \left( \frac { x } { 2 } - 5 \right) ^ { 4 }$

Question

Quizplus · Accepted Answer

To find the local extrema and inflection points, we need to first find the first and second derivative of the function:

$$y = x\left(\frac{x}{2}-5\right)^4$$

$$y' = \left(\frac{x}{2}-5\right)^4 + 4x\left(\frac{x}{2}-5\right)^3$$

$$y'' = 5\left(\frac{x}{2}-5\right)^3 + 12x\left(\frac{x}{2}-5\right)^2$$

Setting the first derivative equal to zero, we get:

$$\left(\frac{x}{2}-5\right)^3 + 4x\left(\frac{x}{2}-5\right)^2 = 0$$

$$\left(\frac{x}{2}-5\right)^2\left[\frac{x}{2}-5 + 4x\right] = 0$$

Solving for $x$, we get $x=0$ or $x=20$.

Now, we need to test the intervals between these two critical points and the endpoints to determine the behavior of the function and the local extrema and inflection points.

Testing $(-\infty,0)$, we can use test points to determine that the function is decreasing and concave up.

Testing $(0,20)$, we can use test points to determine that the function is increasing and concave down between $x=0$ and $x=10$, then increasing and concave up between $x=10$ and $x=20$.

Testing $(20,\infty)$, we can use test points to determine that the function is increasing and concave up.

Therefore, we have a local minimum at $x=0$ and a local maximum at $x=10$, and an inflection point at $x=10$.

The correct graph is Choice B, which shows the local minimum at $(0,0)$, local maximum at $(10,125000)$, and inflection point around $(10,3179)$.

The other choices do not accurately show the location of the local extrema and inflection point.

Sketch the Graph and Show All Local Extrema and Inflection =x(x2−5)4= x \left( \frac { x } { 2 } - 5 \right) ^ { 4 }=x(2x​−5)4

Sketch the Graph and Show All Local Extrema and Inflection $= x \left( \frac { x } { 2 } - 5 \right) ^ { 4 }$