Advanced High-School Mathematics

164 CHAPTER 3 Inequalities

We have

0 = b^2 = 4ac

= m^2 − 4

which immediately givesm =±2. Only the valuem = 2 is rele-

vant here (as we assumed thatx >0) and this is the sought-after

minimum value ofg. (The extraneous valuem=−2 can easily be

seen to be the maximum value ofx+

1

x

, x <0.)

Example 3. Find the equations of

the two straight lines which pass

through the point P(0,5) and are

both tangent to the graph of

y= 4−x^2.

6

y

x

y= 4−x^2





+

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@

@

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Solution. If we write a line with equation`:y= 5 +mx, where the

slope is to be determined, then we are solving 4−x^2 = 5 +mxso

that a double root occurs (i.e., tangency). Clearly, there should

result two values of m for this to happen. Again, the discrimi-

nant is a very good tool. Write the quadratic equation having the

multiple root asx^2 +mx+ 1 = 0, and so

0 = b^2 − 4 ac

= m^2 − 4 ⇒m=± 2.

Therefore, the two lines are given by equations

y= 5 + 2x and y= 5− 2 x.

(The two points of tangency are at the points with coordinates

(± 1 ,3).)