Let f ( x ) be a continues function for all x , and <mrow class="MJX-TeXAtom

Jeramiah Campos 2022-06-25 Answered
Let f ( x ) be a continues function for all x, and | f ( x ) | 7 for all x.

Prove the equation 2 x + f ( x ) = 3 has one solution.

I think the intermediate value theorem is key in this, but I'm not sure of the proper usage.
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Answers (1)

Angelo Murray
Answered 2022-06-26 Author has 23 answers
Let g ( x ) = f ( x ) + 2 x 3. Then g ( 2 ) and g ( 5 ) are...
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Prove that f ( x ) = 3 has a solution on the interval [ a , b ]

And Intermediate Value Theorem says that

if f ( a ) f ( b ) < 0, then it has a solution on that interval

So instead of seeing if 3 is between the interval and stuff like that. Can't I just do this:
f ( x ) = 3
f ( x ) 3 = 0
And then we consider f ( x ) 3 a completely new whole function called g ( x ) = 0

According to the theorem, I can say that since g ( a ) g ( b ) < 0, it has a solution on the interval.

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