Locate all critical points a)(t-5)^2*(t+3)^5 b)(2t-3)^3*(6-2t)^4
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You should remember that critical values represent the solutions to the following equaation such that:
`((t-5)^2*(t+3)^5)' = 0`
You need to use the product rule such that:
`((t-5)^2)'*(t+3)^5 + (t-5)^2*((t+3)^5)' = 0`
`2(t-5)(t+3)+(t-5)^2*(5(t+3)^4) = 0`
You need to factor out `(t-5)(t+3) ` such that:
`(t-5)(t+3)(2 + 5(t-5)(t+3)^3) = 0`
You need to solve the following equations such that:
`{(t-5=0),(t+3=0),(2+5(t-5)(t+3)^3=0):}` `=gt{(t=5),(t=-3),(2+5(t-5)(t+3)(t+3)^2=0):}`
`2 + 5(t^2-2t-15)(t^2+6t+9)=0`
`2+5(t^4+6t^3+9t^2-2t^3-12t^2-18t-15t^2-90t-135)=0`
`2+5t^4+30t^3+45t^2-10t^3-60t^2-90t-75t^2-450t-675 = 0`
`5t^4+20t^3-90t^2-540t-673 = 0`
The equation has two real roots `t=4.9` and `t=-2.6` and two complex roots.
Hence, evaluating the critical values of the function yields `t=-3, t=-2.6, t=4.9` and `t=5` .
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