Answer :
Final answer:
The critical numbers, or the points at which the rate of change of a function is zero, of the function f(x) = 6x⁷ - 7x⁶ are x = 0 and x = 1.
Explanation:
To find the critical numbers of a function, we first need to understand what a critical number is. A critical number of a function is a number where the function's derivative is either zero or undefined.
Therefore, to find the critical number(s) of the function f(x) = 6x⁷ - 7x⁶, we first need to find its derivative.
To find the derivative of f(x), we apply the power rule, which states that the derivative of xⁿ is n*x⁽ⁿ⁻¹⁾.
Using this rule, the derivative of f(x) is f'(x) = 42x⁶ - 42x⁵.
Next, we set this derivative equal to zero to find the x values at which the function's slope (rate of change) is zero, that is, 42x⁶ - 42x⁵ = 0.
Simplify to get x⁶(42 - 42x) = 0.
From this equation, we find that the critical numbers are x = 0 and x = 1.
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