Use the Intermediate Value Theorem to prove that the equation − x 3 + 4 sin ⁡ ( x ) + 4 cos 2 ⁡ ( x ) = 0has at least two solutions. You should carefully justify each of the hypothesis of the theorem.How do I know that at least two exist because I know you're meant to look for a change in sign.

Question

Use the Intermediate Value Theorem to prove that the equation  −      x    3    +  4  sin  ⁡  (  x  )  +  4      cos    2    ⁡  (  x  )  =  0has at least two solutions. You should carefully justify each of the hypothesis of the theorem.How do I know that at least two exist because I know you&#039;re meant to look for a change in sign.

Arturo Wallace · Accepted Answer

There is no problem with continuity. Hence it remains to show two intervals, where the function   f changes sign.   f  (  0  )  =  4,   f  (  2  π  )  =  4  −  (  2  π      )    3    &amp;lt;  0. The next point needs more precise computation:   f  (  −  π      /    2  )  =  (  π      /    2      )    3    −  4  &amp;lt;  0.Hence there exist roots in   (            −      π        2    ,  0  ) and   (  0  ,  2  π  ).We used IVT in the form: If a continuous function has values of opposite sign inside an interval, then it has a root in that interval.

Use the Intermediate Value Theorem to prove that the equation − x

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