The curve $\displaystyle y=hx^2+\frac{k}{x}$ has a turning point at $\displaystyle P(\frac{1}{2},6)$.Find
(i)the value of $\displaystyle h$ and $\displaystyle k$
(ii)the equation of tangent to the curve at $\displaystyle x=1$
the point $\displaystyle P\left(\frac{1}{2} , 6 \right)$ is on the curve ...
$\displaystyle 6 = \frac{h}{4} + 2k$
since $\displaystyle P$ is a turning point ...
$\displaystyle y'\left(\frac{1}{2}\right) = 0$
use the above equation to get a second equation in terms of $\displaystyle h$ and $\displaystyle k$ ... then solve the system of equations.