Increasing and decreasing intervalsFind a polynomial f(x) of degree 4 which increases in the intervals ( − ∞ , 1 ) and (2,3) and decreases in the intervals (1,2) and ( 3 , ∞ ) and satisfies the condition f ( 0 ) = 1.It is evident that the function should be f ( x ) = a x 4 + b x 3 + c x 2 + d x + 1. I differentiated it. Now, I'm lost. I tried putting f ′ ( 1 ) &gt; 0, f ′ ( 3 ) − f ′ ( 2 ) ≥ 0, and f ′ ( 2 ) − f ′ ( 1 ) ≤ 0. Am I doing correct? Or is there another method?

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Increasing and decreasing intervalsFind a polynomial f(x) of degree 4 which increases in the intervals   (  −      ∞    ,  1  ) and (2,3) and decreases in the intervals (1,2) and   (  3  ,      ∞    ) and satisfies the condition   f  (  0  )  =  1.It is evident that the function should be   f  (  x  )  =  a      x    4    +  b      x    3    +  c      x    2    +  d  x  +  1. I differentiated it. Now, I&#039;m lost. I tried putting       f    ′    (  1  )  &amp;gt;  0,       f    ′    (  3  )  −      f    ′    (  2  )  ≥  0, and       f    ′    (  2  )  −      f    ′    (  1  )  ≤  0. Am I doing correct? Or is there another method?

Sanai Douglas · Accepted Answer

Step 1According to the given conditions, we may assume   f  (  x  )  =  a      x    4    +  b      x    3    +  c      x    2    +  d  x  +  1..Then       f    ′    (  x  )  =  4  a      x    3    +  3  b      x    2    +  2  c  x  +  d  ..Notice that   x  =  1  ,  2  ,  3 are the roots for       f    ′    (  x  )  =  0. Hence       f    ′    (  x  )  =  4  a      x    3    +  3  b      x    2    +  2  c  x  +  d  =  k  (  x  −  1  )  (  x  −  2  )  (  x  −  3  )  =  k      x    3    −  6  k      x    2    +  11  k  x  −  6  k  ..Step 2Thus,   a  =      k    4    ,           b  =  −  2  k  ,           c  =      11    2    k  ,           d  =  −  6  k  .Therefore, we obtain   f  (  x  )  =      k    4        x    4    −  2  k      x    3    +            11      k        2        x    2    −  6  k  x  +  1  ,, where   k  &amp;gt;  0..

Makayla Eaton · Accepted Answer

Step 1Since 1,2,3 are zeroes for f′ we have       f    ′    (  x  )  =  a  (  x  −  1  )  (  x  −  2  )  (  x  −  3  )  =  a      x    3    −  6  a      x    2    +  11  a  x  −  6  a.So, by integrating we get   f  (  x  )  =            a      4            x    4    −  2  a      x    3    −                    11        a            2            x    2    −  6  a  x  +  cStep 2Puting   f  (  0  )  =  1 we get   c  =  1 and thus:  f  (  x  )  =            a      4            x    4    −  2  a      x    3    −                    11        a            2            x    2    −  6  a  x  +  1

Find a polynomial f(x) of degree 4 which increases in the intervals (-infty, 1) and (2,3) and decreases in the intervals (1, 2) and (3, infty) and satisfies the condition f(0)=1.

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