Let f (x)= ax^3 + 6x^2 +bx + 4. Determine the constants a and b so f has a relative minimum at x= -1 and a relative maximum x= 2.

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To find the relative minimum and maximum of the equation `ax^3 + 6x^2 + bx + 4` we take the first derivative,` `

`3ax^2 + 12x + b`

Then substitute the value of relative minimum `x = -1`

`3a (-1)^2 + 12(-1) + b`

`3a - 12 + b` 

`3a + b = 12`  `larr` Equation 1

And substitute the value of relative maximum `x = 2`

`3a (2)^2 + 12(2) + b`

`12a + 24 + b`

`12 a + b = -24` `larr` Equation 2

So, we have two equations

`3a + b = 12`

`12a + b = -24`

Using elimination method, to solve for `a`

`-3a - b = -12` 

`12a + b = -24`

`9a = -36`

`a = -4`

Then using the value `a = - 4` ,we substitute it in Equation 1

`3(-4) + b = 12`

`-12 + b = 12`

`b = 24`

Thus, the value of `a` is `-4` and `b` is `24`

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