sign analysis diagram and list the values of x where f(x) has local maximums and local minimums. Suppose that f'(x)= (2x^2 -5x-12)/(3x^2 -7x+2) and that f(x) is continuous for all real numbers x. Do a sign analysis diagram and list the values of x where f(x) has local maximums and local minimums. In addition, list the intervals where f(x) is increasing and decreasing.

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You need to evaluate the roots of equation f'(x) = 0 such that:

`(2x^2 -5x-12)/(3x^2 -7x+2) = 0 =gt 2x^2 -5x-12 = 0`

You should use quadratic formula such that:

`x_(1,2) =(5+-sqrt(25+96))/4`

`x_(1,2) =(5+-sqrt121)/4 =gt x_(1,2) =(5+-11)/4`

`x_1 = 4 ; x_2 = -6/4 = -3/2`

You should remember the law of signs of quadratic, hence, f'(x) has negative values between `(-3/2,4) ` and it has positive values in `(-oo,-3/2) ` and `(4,oo).`

The function increases in `(-3/2,4) ` and it decreases in  (-oo,-3/2) and `(4,oo).`

The function reaches its minimum at `x = -3/2`  and its maximum at `x=4.`

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