+76 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!
+118609
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
