1.
find the values of m for which the line y=mx-3 is a tangent to the curve y=x+(1/x) and find the x-coordinate of the point at which this tangent touches the curve.

2.
find the value of c and d for which {x: -5<x<3} is the solution set of
x^2 + cx < d .

2. Originally Posted by wintersoltice
1.
find the values of m for which the line y=mx-3 is a tangent to the curve y=x+(1/x) and find the x-coordinate of the point at which this tangent touches the curve.
[snip]
Consider the tangent at (a, a + 1/a):

$\frac{dy}{dx} = 1 - \frac{1}{x^2} \Rightarrow m = 1 - \frac{1}{a^2}$.

Therefore the equation of the tangent is:

$y - \left(a + \frac{1}{a} \right) = \left(1 - \frac{1}{a^2} \right) (x - a) \Rightarrow y = \left(1 - \frac{1}{a^2} \right) x + \frac{2}{a}$.

Therefore $\frac{2}{a} = -3 \Rightarrow a = - \frac{2}{3}$ ......

3. Originally Posted by wintersoltice
[snip]2.
find the value of c and d for which {x: -5<x<3} is the solution set of
x^2 + cx < d .
$x^2 + cx - d < 0$.

Therefore $x^2 + cx - d = (x + 5)(x - 3) = x^2 + 2x - 15$ .....