Use the first derivative to determine where the function f(x)=6x^4+192x is increasing and decreasing. hint: X^3+a^3=(x+a)(x^2-ax+a^2) increasing: decreasing:

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pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

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Function is an increasing function if



`x+2>0 and x^2+4-2x>0`

since f is real function so ,only possible `x>2`

Thus function is increasing if `x in(2,oo)` and decreasing is `x in(-oo,2)`

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oldnick | (Level 1) Valedictorian

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`f(x)=6x^4+192x=6x(x^3-32)`   (1)

`f'(x)= 24x^3+192`

`f'(x)=24x^3+192=0`     dividing by 24:

`x^3+8=0`      then    `x=-2` 

`f''(x)=3x^2`    and `f''(-2)=12 >0`  

Thus `x= -2`  is a min. point, it means function decrease  for `x< -2`    and decrease for `x>2`   assuming the value  `y=-288`  for  `x=-2`


by the  (1) we know has zero for `x= 0`   and  `x=-2 ^3sqrt(4)`