Question

As you approach the seventh gate, you notice that it is only a short distance to the area where your computer indicated the communication device was stored. You look past the gate hopefully, as your computer translates: The passcode for this gate is found by determining the domain of the derivative of f(x) and then evaluating the derivative at x= -1/34 Enter the domain in interval notation.

f(x)= √8−34x
Domain of f'(x)=__

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Answer 1

The passcode for the seventh gate is found by determining the domain of the derivative of f(x) and evaluating the derivative at x = -1/34. The domain of f'(x) is (-∞, 4/17) and the value of f'(-1/34) is -17/3.

Step-by-step explanation:

The passcode for the seventh gate can be found by determining the domain of the derivative of f(x) and evaluating the derivative at x = -1/34. Let's find the derivative of f(x) first:

Given f(x) = √(8 - 34x), we can rewrite it as f(x) = (8 - 34x)^(1/2).

Now, let's find the derivative of f(x) using the power rule:

f'(x) = (1/2)(8 - 34x)^(-1/2)(-34) = -17(8 - 34x)^(-1/2).

The domain of f'(x) is the set of all x-values for which the derivative is defined. Since (8 - 34x)^(-1/2) is defined as long as 8 - 34x > 0, we need to solve the inequality:

8 - 34x > 0

Solving for x, we get x < 8/34 = 4/17.

Therefore, the domain of f'(x) is (-∞, 4/17).

To find the passcode, we need to evaluate the derivative at x = -1/34:

f'(-1/34) = -17(8 - 34(-1/34))^(-1/2) = -17(9)^(-1/2) = -17/3.

So, the passcode for the seventh gate is -17/3.