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!
Please just ask one question per post.
You would probably be best off to repost the second one
I'll answer the first one.
If f''' = 0
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