Consider the function $f ( x ) = \frac { 1 } { x ^ { 2 } - k x }$, where $k$ is a nonzero constant. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { k - 2 x } { \left( x ^ { 2 } - k x \right) ^ { 2 } }$. (a) Let $k = 3$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 3 x }$. Write an equation for the line tangent to the graph of $f$ at the point whose $x$-coordinate is 4. (b) Let $k = 4$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 4 x }$. Determine whether $f$ has a relative minimum, a relative maximum, or neither at $x = 2$. Justify your answer. (c) Find the value of $k$ for which $f$ has a critical point at $x = -5$. (d) Let $k = 6$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 6 x }$. Find the partial fraction decomposition for the function $f$. Find $\int f ( x ) \, dx$.
Consider the function $f ( x ) = \frac { 1 } { x ^ { 2 } - k x }$, where $k$ is a nonzero constant. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { k - 2 x } { \left( x ^ { 2 } - k x \right) ^ { 2 } }$.
(a) Let $k = 3$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 3 x }$. Write an equation for the line tangent to the graph of $f$ at the point whose $x$-coordinate is 4.
(b) Let $k = 4$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 4 x }$. Determine whether $f$ has a relative minimum, a relative maximum, or neither at $x = 2$. Justify your answer.
(c) Find the value of $k$ for which $f$ has a critical point at $x = -5$.
(d) Let $k = 6$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 6 x }$. Find the partial fraction decomposition for the function $f$. Find $\int f ( x ) \, dx$.