How to solve this problem

Tenor

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Mar 5, 2023
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At which points on the graph of inverse of f(x)=1x2+1+12x3f(x)= \frac 1 {x^2+1} +\sqrt[3]{1-2x} x0x \ge 0 the tangents of f(x) and its inverse are perpendicular?
 
 
Hello. Where do you need help, in this exercise? Please share how far you got, so that people can get a sense of which parts you already understand. For example, have you learned the concept of inverse functions and how to find them?
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At which points on the graph of inverse of f(x)=1x2+1+12x3f(x)= \frac 1 {x^2+1} +\sqrt[3]{1-2x} x0x \ge 0 the tangents of f(x) and its inverse are perpendicular?

Finding an explicit expression for f1(x)f^{-1}(x) algebraically is going to be quite difficult. The best that I could do to find a perpendicular pair of tangent lines was to look at the graphs of the function and its inverse. Given the restriction that x0x \ge 0, I can see one possible pair.

inverse_perp.jpg
 
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