Functions 1996 AHSME #25

Suppose that . What is the maximum value of ?

MAIN CONCEPTS
Geometrically analyze and
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Before we proceed, I would like to mention that this is a difficult functions question and it requires some deep analytical skills to answer. Let’s try our best and see what solution we can come up with.

From inspection, we first notice that the equation  will actually give us a circle. Geometrically we are able to graph out the equation. Thus the question is asking us to pick a point on the circle to maximize . Here lies the main crux of the problem. We are to maximize not , but . Knowing that x and y varies, it is already hard enough to find a maximum of the two values added up let alone attaching constants 3 and 4 to them. A use of calculus of variations would be helpful but fortunately we can proceed using algebraic methods.

Let us first complete the square.

giving us

We know that this gives us a circle of radius 8 and centre (7,3). Our job is to pick a point of the circle so as to maximize  where x and y are the respective coordinates of the circle.

How about we take it from the  expression. Suppose I let this equal to a variable, that is

and rearranging, I get

I can now graph the above equation and attempt to maximize c. With some knowledge of Cartesian formulas, i see that this is a line of slope  and y-intercept . So, consider geometrically, we need to find the line with slope  that intersects the circle and has the largest value for its y-intercept. The line is as shown below.

The line will be tangent to the curve and the line segment that is the radius to this tangent point has slope . Here comes the second hurdle. While you may think you already have the answer, there is actually some difficulty solving for x and y. With the usual methods, we will substitute

into the equation of the circle, but notice two things. There are two unknowns c and x, remember we have yet to find c, and that we will most probably end with a equation with  and  which will be messy to solve.

Instead, with some geometric intuition, notice that I can express the point as . I am basically taking reference from the centre and using the knowledge of

where  is the normal to the line and this normal connects the centre to the point. In this case  giving the ratio of the horizontal and vertical components as 3 : 4 and so we now conveniently solve for a.

Substituting  back into the equation of the circle gives me

So the point that gives the maximum value of  is  or  which yields the maximum value of

This is certainly not an easy question. Nonetheless, with some geometric inspection, we can work our way to a solution.