June 4th, 2015, 05:25 AM  #1 
Newbie Joined: May 2015 From: sweden Posts: 12 Thanks: 0  prove function invertible
Hi! i need help solving this problem. Prove that f(x)= x^7+5x^3+3 is invertible and find the derivative to the inverse function at the point 9 Im not really sure how to do this. i know it has to be injective to be invertible but i dont know how to prove it with this function, and if it it the way i should do it? please help 
June 4th, 2015, 05:34 AM  #2 
Senior Member Joined: Mar 2011 From: Chicago, IL Posts: 214 Thanks: 77 
f(x) is continuous. The derivative of the function $\displaystyle f'(x)=7x^6+15x^2$ is always positive. It means that f(x) is a monotonic function. Hence it is invertible. 
June 4th, 2015, 05:44 AM  #3  
Newbie Joined: May 2015 From: sweden Posts: 12 Thanks: 0  Quote:
Do i need to find the inverse function? because i tried doing that and it didnt really work  
June 4th, 2015, 05:56 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
If g(x) is the inverse function to f(x) then f(g(x))= x. By the chain rule, f'(g(x))g'(x)= 1 so that g'(x)= 1/f'(g(x)). The derivative of g(x) at x= 9 is 1 over the derivative of f at the x value such that f(x)= 9. What is x there? (Hint it's easy!)

June 4th, 2015, 06:05 AM  #5  
Newbie Joined: May 2015 From: sweden Posts: 12 Thanks: 0  Quote:
Derivative of g(x) is 1/ the derivative of f(1)? or did i understand wrong? Because the derivative of that would just be zero...?  
June 4th, 2015, 06:40 AM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra  Quote:
Quote:
 
June 4th, 2015, 08:03 AM  #7 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  

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