Thread: Two problems dealing with linear approximation and inflection points

Okay here are two problems that have my friends and I stumped as we can't figure out a correct answer.

Problem #1:
At the point where x=3 on the curve y = (ax+1)/(x-2) , where a is a constant the slop of the normal is 1/5. Find the value of a.

Its multiple choice and the answers they give us are 1/2, 2, 0, 3, 1/3.
However, I keep getting -5/3 for a.


Problem #2:
The function y=x^4 - 4x^3 + 14 has a horizontal inflection at (A, B). The value of A+B is what?

Yet again it is another multiple choice question. The answers given are -10, 0, -39, 14, or none of these. I have a question though. In class we talk about points of inflection but in this question it talks about horizontal inflection. Is there a difference? We haven't touched base at all in class if it is different.
We found two points of inflection and they were (-2, 2) and (0,14) but we don't know which one is the correct answer if any. Its either 0 or 14 or maybe not if we screwed up in our calculations.

Any help would be appreciated.

Originally Posted by forkball42

Problem #1:
At the point where x=3 on the curve y = (ax+1)/(x-2) , where a is a constant the slop of the normal is 1/5. Find the value of a.

Its multiple choice and the answers they give us are 1/2, 2, 0, 3, 1/3.
However, I keep getting -5/3 for a.

The slope of the tangent to the curve can be found by






So at x = 3, the tangent to the curve has the value


Now, the normal to the curve at this point is a line with a slope perpendicular to the tangent line. So the slope of the normal line is


So solving this I get that a = 2.

-Dan

For #1 the slope is given by

The normal has slope 1/5, therefore, the slope of the tangent at x=3 must have slope -5.

So, when x=3,

Solving for a we see a=2

Originally Posted by forkball42

Problem #2:
The function y=x^4 - 4x^3 + 14 has a horizontal inflection at (A, B). The value of A+B is what?

Yet again it is another multiple choice question. The answers given are -10, 0, -39, 14, or none of these. I have a question though. In class we talk about points of inflection but in this question it talks about horizontal inflection. Is there a difference? We haven't touched base at all in class if it is different.
We found two points of inflection and they were (-2, 2) and (0,14) but we don't know which one is the correct answer if any. Its either 0 or 14 or maybe not if we screwed up in our calculations.

This is my best guess after doing a search: A "horizontal inflection point" likely what you would simply call an inflection point. Now, recall that a critical point exists when the first derivative is either 0 or undefined. If the critical point is one where the first derivative is undefined and the second derivative at this point is also undefined, we have what you might call a "vertical inflection point."

In any event, all the inflection points here have a horizontal "tangent" so I would imagine they fall under the category of horizontal inflection points.





So we have critical points at


The second derivative is






So I get that there is only one inflection point for the function: at x = 0. The y value at x = 0 is y = 14. So the point (A, B) = (0, 14) which implies that A + B = 0 + 14 = 14.

-Dan