Question on Local Maxima and Local MinimaFind the set of all the possible values of a for which the function f ( x ) = 5 + ( a − 2 ) x + ( a − 1 ) x 2 − x 3 has a local minimum value at some x &lt; 1 and local maximum value at some x &gt; 1.The first derivative of f(x) is: f ′ ( x ) = ( a − 2 ) + 2 x ( a − 1 ) − 3 x 2 I do know the first derivative test for local maxima and local minima, but I can't figure out how I could use monotonicity to find intervals of increase and decrease of f′(x)The expression for f′(x) might suggest the double derivative test is the key, considering f ″ ( x ) = 2 ( a − 1 ) − 6 x for which the intervals where it is greater than zero and less than zero can be easily found, but then again I can't think of a way how I could find a c such that f ′ ( c ) = 0.

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Question on Local Maxima and Local MinimaFind the set of all the possible values of a for which the function   f  (  x  )  =  5  +  (  a  −  2  )  x  +  (  a  −  1  )      x    2    −      x    3   has a local minimum value at some   x  &amp;lt;  1 and local maximum value at some   x  &amp;gt;  1.The first derivative of f(x) is:      f    ′    (  x  )  =  (  a  −  2  )  +  2  x  (  a  −  1  )  −  3      x    2  I do know the first derivative test for local maxima and local minima, but I can&#039;t figure out how I could use monotonicity to find intervals of increase and decrease of f′(x)The expression for f′(x) might suggest the double derivative test is the key, considering       f    ″    (  x  )  =  2  (  a  −  1  )  −  6  x for which the intervals where it is greater than zero and less than zero can be easily found, but then again I can&#039;t think of a way how I could find a c such that       f    ′    (  c  )  =  0.

autarhie6i · Accepted Answer

Step 1      f    ′    (  x  )  =  0 is an inverse parabola, which has real roots in x if   4  [  (  a  −  1      )    2    +  3  (  a  −  2  )  ]  ≥  0  ,, or,   a  ∈      [                  −        1        −                  21                    2        ,                  −        1        +                  21                    2        ]    ..Step 2Let that be the case, then       f    ′    (  x  )  =  0 means that there are two critical points, namely      x    1    =            a      −      1      +              (        a        −        1                  )          2                +        3        (        a        −        2        )              3    ,      x    2    =            a      −      1      −              (        a        −        1                  )          2                +        3        (        a        −        2        )              3    .Now, you need to check whether       f    ″    (      x    1    )  ≶  0 and       f    ″    (      x    2    )  ≶  0 as well as whether the critical points satisfy   x  &amp;lt;  1 or   x  &amp;gt;  1 as in your question.

Find the set of all the possible values of a for which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum value at some x<1 and local maximum value at some x>1.

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