Determine the x-coordinate(s) of any relative extrema and inflection points of the function $y = x ^ { 5 } \ln \frac { x } { 9 }$
A)relative minimum: $x = 9 e ^ { \frac { - 1 } { 5 } }$; no inflection points
B)relative maximum: $x = 9 e ^ { \frac { - 1 } { 5 } } ;$ inflection point: $x = 9 e ^ { \frac { - 9 } { 20 } }$
C)no relative maximum or minimum; inflection point: $x = 9 e ^ { \frac { - 1 } { 5 } }$
D) no relative extrema or inflection points.
E)relative minimum: $x = 9 e ^ { \frac { - 1 } { 5 } } ;$ inflection point: $x = 9 e ^ { \frac { - 9 } { 20 } }$

Question

Determine the x-coordinate(s) of any relative extrema and inflection points of the function $y = x ^ { 5 } \ln \frac { x } { 9 }$ 
A)relative minimum: $x = 9 e ^ { \frac { - 1 } { 5 } }$; no inflection points 
B)relative maximum: $x = 9 e ^ { \frac { - 1 } { 5 } } ;$ inflection point: $x = 9 e ^ { \frac { - 9 } { 20 } }$
C)no relative maximum or minimum; inflection point: $x = 9 e ^ { \frac { - 1 } { 5 } }$
D) no relative extrema or inflection points.
E)relative minimum: $x = 9 e ^ { \frac { - 1 } { 5 } } ;$ inflection point: $x = 9 e ^ { \frac { - 9 } { 20 } }$

Quizplus · Accepted Answer

To find relative extrema and inflection points, we differentiate the function and set the derivatives equal to zero. The first derivative gives us critical points, which can indicate relative extrema, and the second derivative gives us points of inflection. For the given function $y = x^5 \ln\left(\frac{x}{9}\right)$, the first derivative will involve both $x^5$ and the natural logarithm function, and setting it to zero will help identify any critical points. The second derivative will help determine concavity changes, indicating inflection points. The specific values given in option E for a relative minimum and an inflection point are consistent with the behavior of such a function, where the logarithmic component influences the location of extrema and inflection points based on its properties and interaction with the polynomial term.

Determine the X-Coordinate(s) of Any Relative Extrema and Inflection Points y=x5ln⁡x9y = x ^ { 5 } \ln \frac { x } { 9 }y=x5ln9x​

Determine the X-Coordinate(s) of Any Relative Extrema and Inflection Points $y = x ^ { 5 } \ln \frac { x } { 9 }$