Answer :
Final answer:
The critical numbers for the function f(x)=12x⁵+60x⁴ −240x³ +7 are found by finding the derivative of the function and setting it equal to zero. The critical numbers are the solutions of this equation, which are x=0, x=2, and x=-6. Therefore, A, which is the smallest x value that represents the lower limit of our function, is -6.
Explanation:
To find the critical numbers of the function f(x)=12x⁵+60x⁴ −240x³ +7, we first need to find the derivative of the function (f'(x)) and then set it equal to zero and solve for x.
The derivative of the function is f'(x)=60x⁴+240x³-720x². Setting f'(x)=0 gives us: 60x⁴+240x³-720x²=0.
Factoring out 60x², we get 60x²(x²+4x-12)=0. Setting this equal to zero gives us the solutions x=0, x=2, and x=-6. These are the critical points of f(x).
The interval A is the smallest x value that represents the lower limit of our function, in this case, A would be -6.
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