High School

Find the critical number(s), if any, of the function.

\[ f(x) = 6x^{7} - 7x^{6} \]

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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