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!

yeliah Nov 2, 2020

#3**+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

Melody Nov 3, 2020