suppose we have following function f ( x ) = x 2 + 10 sin ⁡ ( x )we should show that there exist number c such that f ( c ) = 1000clearly we can solve this problem using intermediate value theorem for instance f ( 0 ) = 0 f ( 90 ) = 90 2 + 10 sin ⁡ ( 90 ) = 8100 + 10 ∗ 1 = 8110and because 1000 is between these two number we can see that there exist such c so that f ( c ) = 1000am i right? thanks in advance

Question

suppose we have following function  f  (  x  )  =      x    2    +  10  sin  ⁡  (  x  )we should show that there exist number   c such that   f  (  c  )  =  1000clearly we can solve this problem using intermediate value theorem for instance  f  (  0  )  =  0  f  (  90  )  =      90    2    +  10  sin  ⁡  (  90  )  =  8100  +  10  ∗  1  =  8110and because 1000 is between these two number we can see that there exist such c so that   f  (  c  )  =  1000am i right? thanks in advance

Trey Harrington · Accepted Answer

f    ′    (  x  )  =  2  x  +  10  cos  ⁡  x is clearly positive if   x  ≥  6, hence our function is increasing over the interval   [  6  ,  33  ] and since 1000 is between   f  (  6  ) and   f  (  33  ), there is some (only one)   c  ∈  (  6  ,  33  ) such that   f  (  c  )  =  1000.

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