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1. We can repeat the derivative operation: the second derivative f'' is the derivative of f', the third derivative f''' is the derivative of f'' , and so on. Find a function f such that f' and f'' are not the function 0, but f''' is the constant function 0. 

 

2. 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).

 

I am unsure how to do these. Will someone please help? Thanks!

 Nov 2, 2020
 #1
avatar+111546 
0

Please just ask one question per post.

 Nov 2, 2020
 #2
avatar+76 
0

sorry, should I repost?

yeliah  Nov 2, 2020
 #3
avatar+111546 
+2

 

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

 Nov 3, 2020
 #4
avatar+76 
+1

thank you so much!

I understand now!

yeliah  Nov 3, 2020
edited by yeliah  Nov 3, 2020

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