Find the Domain (dk)/(ds)=(-4s^3-10s+54)/((s^2-6s+16)^2)

Problem

d(k)/d(s())=(−4*s3−10*s+54)/((s2−6*s+16)2)

Solution

1. Identify the conditions for the domain of a rational function. The domain consists of all real numbers except those that make the denominator equal to zero.

2. Set the denominator equal to zero to find the excluded values:

(s2−6*s+16)2=0

3. Simplify the equation by taking the square root of both sides:

s2−6*s+16=0

4. Apply the quadratic formula s=(−b±√(,b2−4*a*c))/(2*a) to solve for s() where a=1 b=−6 and c=16

s=(−(−6)±√(,(−6)2−4*(1)*(16)))/(2*(1))

5. Calculate the discriminant to determine the nature of the roots:

Discriminant=36−64

Discriminant=−28

6. Conclude that since the discriminant is negative, there are no real values of s() that make the denominator zero. Therefore, the denominator is always positive for all real numbers.

Final Answer

Domain=(−∞,∞)

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