Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 8 } } { 8 } - \ln x$ . Use a graphing utility to confirm your results.
A)relative maximum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; inflection point at x = 0
B)relative minimum value $y = 8$ at $x = 1$ ; inflection point at x = 0
C)relative minimum value $y = 8$ at $x = 1$ ; no inflection points
D)relative minimum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; no inflection points
E)relative maximum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; no inflection points

Question

Quizplus · Accepted Answer

To find relative extrema, we first find the derivative of $y = \frac { x ^ { 8 } } { 8 } - \ln x$, which is $y' = x^7 - \frac{1}{x}$. Setting $y'$ equal to zero gives $x^7 - \frac{1}{x} = 0$, which simplifies to $x^8 = 1$, so $x = 1$. To determine if this is a maximum or minimum, we can use the second derivative test or simply note that for $x < 1$, $y'$ is negative, and for $x > 1$, $y'$ is positive, indicating a minimum at $x = 1$. The value of $y$ at $x = 1$ is $\frac{1^8}{8} - \ln(1) = \frac{1}{8}$. To find inflection points, we find the second derivative, $y'' = 7x^6 + \frac{1}{x^2}$, and set it equal to zero. However, $y''$ is always positive for $x > 0$, indicating there are no points where the concavity changes, hence no inflection points. Therefore, the correct answer is that there is a relative minimum value of $y = \frac{1}{8}$ at $x = 1$, with no inflection points, matching option D.

Locate Any Relative Extrema and Inflection Points of the Function y=x88−ln⁡xy = \frac { x ^ { 8 } } { 8 } - \ln xy=8x8​−lnx

Locate Any Relative Extrema and Inflection Points of the Function $y = \frac { x ^ { 8 } } { 8 } - \ln x$