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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).

Nov 2, 2020

#1
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Nov 2, 2020
#2
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sorry, should I repost?

yeliah  Nov 2, 2020
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You would probably be best off to repost the second 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
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thank you so much!

I understand now!

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