Let f(x) = (x − 3)^−2.Find all values of c in (1, 4)

such that f(4) − f(1) = f'(c)(4 − 1).(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

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i NEED A LIST OF THE LEAST POPULAR STATe/ NATINoal parks

To find all values of c in the interval (1, 4) such that f(4) − f(1) = f'(c)(4 − 1), we first need to find the values of f(4), f(1), and f'(x).

Given that f(x) = (x − 3)^−2, we can find f(4) and f(1) by substituting the respective values of x into the function:
f(4) = (4 − 3)^−2 = (1)^−2 = 1
f(1) = (1 − 3)^−2 = (-2)^−2 = 1/4

To find f'(x), which is the derivative of f(x), we can differentiate the function with respect to x:
f'(x) = d/dx (x − 3)^−2

To differentiate this function, we can use the chain rule. Let u = x − 3, then we have:
f'(x) = d/du u^−2 * du/dx

Taking the derivatives separately, we have:
d/du u^−2 = -2u^−3
du/dx = 1

Substituting these back into f'(x), we get:
f'(x) = -2(x − 3)^−3

Now, we can substitute the values of f(4), f(1), and f'(x) into the equation f(4) − f(1) = f'(c)(4 − 1):

1 − 1/4 = -2(c − 3)^−3 * 3

Simplifying this equation, we have:
3/4 = -6(c − 3)^−3

To find all values of c that satisfy this equation, we can solve for (c − 3)^−3:

(c − 3)^−3 = -3/8

To get rid of the negative sign, we can take the cube root of both sides:

c − 3 = (−3/8)^(1/3)

To solve for c, we can add 3 to both sides:

c = 3 + (−3/8)^(1/3)

Therefore, the value of c in the interval (1, 4) that satisfies the equation f(4) − f(1) = f'(c)(4 − 1) is given by c = 3 + (−3/8)^(1/3).