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Answer :
The critical numbers of the function f(x)=12x^5+45x^4-100x^3+7 are found by setting its derivative to zero. The critical numbers are x=0, x=1, and x=-5.
To find the critical numbers for f(x) = 12x5 + 45x4 - 100x3 + 7, we first need to calculate the derivative f'(x).
f'(x) = 60x4 + 180x3 - 300x2. Setting this equal to zero gives us 60x4 + 180x3 - 300x2 = 0.
Factoring out the common terms, we get x2(60x2 + 180x - 300) = 0, which simplifies to x2(x - 1)(60x + 300) = 0. We find that x = 0, x = 1, and x = -5 are the critical numbers. Classification involves checking these values in the second derivative, but as the question did not ask for this, we stop at finding the critical points.
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