Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 15 } } { 15 } - \ln x$ . Use a graphing utility to confirm your results.
A) relative minimum value: $\left( 1 , \frac { 1 } { 15 } \right)$; no inflection points
B) relative minimum value: $( 1,15 )$; inflection point: $( 0,0 )$
C) relative maximum value: $( 1,15 )$; inflection point: $( 0,0 )$
D) relative minimum value: $( 1,15 )$; no inflection points
E) relative maximum value: $\left( 1 , \frac { 1 } { 15 } \right)$; no inflection points

Question

Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 15 } } { 15 } - \ln x$ . Use a graphing utility to confirm your results. 
A) relative minimum value: $\left( 1 , \frac { 1 } { 15 } \right)$; no inflection points
B) relative minimum value: $( 1,15 )$; inflection point: $( 0,0 )$
C) relative maximum value: $( 1,15 )$; inflection point: $( 0,0 )$
D) relative minimum value: $( 1,15 )$; no inflection points
E) relative maximum value: $\left( 1 , \frac { 1 } { 15 } \right)$; no inflection points

Quizplus · Accepted Answer

To find relative extrema, we first find the derivative of $y = \frac { x ^ { 15 } } { 15 } - \ln x$ and set it equal to zero. The derivative is $y' = x^{14} - \frac{1}{x}$. Setting $y' = 0$ gives $x^{14} = \frac{1}{x}$, which simplifies to $x^{15} = 1$, so $x = 1$. To determine if this is a maximum or minimum, we can use the second derivative test. The second derivative is $y'' = 14x^{13} + \frac{1}{x^2}$, which is positive for all $x > 0$, indicating a minimum at $x = 1$. Substituting $x = 1$ into the original function gives $y = \frac{1}{15} - \ln(1) = \frac{1}{15}$, so the relative minimum is at $\left( 1 , \frac { 1 } { 15 } \right)$.To find inflection points, we look for where the second derivative changes sign, but since $y'' = 14x^{13} + \frac{1}{x^2}$ is always positive for $x > 0$, there are no inflection points.

Locate Any Relative Extrema and Inflection Points of the Function y=x1515−ln⁡xy = \frac { x ^ { 15 } } { 15 } - \ln xy=15x15​−lnx

Locate Any Relative Extrema and Inflection Points of the Function $y = \frac { x ^ { 15 } } { 15 } - \ln x$