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Answer :
Final answer:
The critical values of the function f(x) = 2x⁶ - 7x⁵ are 0 and 35/12. Their average is 35/24.
Explanation:
To find all the critical values of the function f(x) = 2x⁶ - 7x⁵, we must first find the derivative of the function, which will yield f'(x) = 12x⁵ - 35x⁴. The critical points are the values of x where the derivative equals zero or is undefined. Setting f'(x) equal to zero gives us the critical points after factoring:
f'(x) = x⁴(12x - 35)
This expression will be zero when x = 0 or when 12x - 35 = 0. Solving the latter gives us x = 35/12. Therefore, the critical values are 0 and 35/12. To compute their average, we add the critical values together and divide by the total number of critical values.
Average = (0 + 35/12) / 2 = 35/24
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