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Answer :
Final answer:
The critical values of f(x) = 2x⁶ - 7x⁵ are 0 and approximately 2.91667. The average of these critical values is approximately 1.45833.
Explanation:
To find the critical values of the function f(x) = 2x⁶ - 7x⁵, we must first find the derivative of the function, set it equal to zero, and solve for x. The derivative of f(x) is f'(x) = 12x⁵ - 35x⁴. Setting this equal to zero gives us:
- 12x⁵ - 35x⁴ = 0
- x⁴(12x - 35) = 0
- x = 0 or x = 35/12
So, the critical values are 0 and 35/12 (approximately 2.91667). To find the average of these critical values:
- (0 + 35/12) / 2 = 35/24 ≈ 1.45833
Therefore, the average of the critical values is approximately 1.45833.
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