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  • one year ago

Prove that 2x-1 and e^(-x) intersect on the interval [0,1].

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  1. SolomonZelman
    • one year ago
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    A good calculus I approach without graphing the functions. (Be careful not to lose me, I will try my best to be clear...) `------------------- START FROM ------------------` I know \(f(x)=2x-1\) and \(g(x)=e^{-x}\) are both continuous functions over the interval of (−∞,+∞) and certainly over the interval of [0,1]. (NOTE: I`m defining your functions as g and f for convenience) `-------- ADDITIONAL PART (continuity of f and g) --------` Will you give me this continuity? If not, then here is a flashback: A polynomial function f, will be continuous over (−∞,+∞), as you know, because it is defined at every value of x. That is by definition. ((Or try to come up with a value of x in any polynomial you choose to disprove this statement. xD )) The exponential g, is also continuous everywhere. \(g(x)=e^{-x}=1/e^x\) (and e^x will never =0 for all x) So.... Since both functions are defined for all real x, they are therefore continuous (−∞,+∞), and thus these both functions are certainly continous over [0,1]. `------------- The essence of the PROVE -------------` Since f and g are both continuous, therefore the range of each function is between it's maximum and minimum. (I assume you get why) If the ranges of f and g over the interval [0,1] intersect, \(\rm \color{red}{ and}\) \(\bf \color{red}{ if~one~function~is~increasing~and~another~is~decreasing}\) (but not necessarily of both decrease or both increase) then, if one increases (in this case function f with a positive slope) and the other decreases (in this case g - b/c it is an exponential decay) then you know the functions will intersect (on that interval). `---------------- HELPFUL NOTE-----------------` I hope I was clear and elaborate enough to follow. I`ll give you another note for the completion. Now, all you need to do is to use calculus to find the absolute maximum and absolute minimum values of f and g over the interval [0,1]. Say you find that: maximum of f is A maximum of g is C minimum of f is B minimum of g is D then the range of f is [A,B] and range of g is [C,D] if these two ranges intersect, then the functions g and f also intersect. Hope this is a helpful feedback for you. Enjoy!

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