Melody

Hi asinus,

I did not see your edit till a moment ago.

your   g'(x) is now correct  but you have not integrated it to get g

What did you get for g ?

What did you plot, if you provided me with a link to your plot I could comment a lot better.

Note: Asinus's answer is not correct.   (Although it is better than before)

\(g'(x)=\frac{-1}{2x}\qquad \text{That is true}\)

But what do you get when you integrate that?

Rubbish Pangolin,

Guest gave you some good advice.

Draw it and count!

You have made an error asinus, can you find it?

Here is another way, (though Heron's formula is probably easier.

Let X be the point of intersection of the circle and the tangent

Let r be the radius

Now you have triangles BCX  and  BAX

The angles at X are right angles so you can use pythagoras's theorum to find XC  and  XA  in terms of r

added they equal 12

solve

(it is a bit tricky)

Let f(x) and g(x) be functions with domain \((0,\infty)\). Suppose \(f(x)=x^2\)  and the tangent line to f(x) at x=a is perpendicular to the tangent line to g(x) at x=a for all positive real numbers a. Find all possible functions g(x).

ok you already have said that the GRADIENT OF THE TANGENT TO F(X) is

f'(x)=2x

So the gradient of the tangent to g(x) at any given point must be

   \(g'(x)= \frac{-1}{2x}\)

So you have to integrate that to get the value of   g(x)

Can you do that?

Do you completely understand ?

When you get an answer plot f and g on desmos.   Check you understand what happens.

Start by dividing top and bottom by 2x

and then simplifying of course.

then show us what you get.

You would probably be best off to repost the second one

I'll answer the first one.

If   f'''  = 0 

then

f'' =k  \

k can be any6 constant but 6 is as good as another

f' = 6x+t     

t can be any constant but -8 is as good as any other

f' = 6x-8

f=   3x^2 -8x + q

q can be any constant but 1 is as good as any other

f=   3x^2 -8x + 1

So there is one.

In fact any quadratic f function would work

Please just ask one question per post.

Have you written out the beginning of the sequence of yourself?

What did you do yourself before you asked for help?