SOLUTION: F(x) =1/x^2 +18x+84 a) give the domain of f in interval novation b) find the critical numbers c) determine intervals where f is increasing and decreasing Use the derivative

Algebra ->  Probability-and-statistics -> SOLUTION: F(x) =1/x^2 +18x+84 a) give the domain of f in interval novation b) find the critical numbers c) determine intervals where f is increasing and decreasing Use the derivative       Log On


   

Question 1180906: F(x) =1/x^2 +18x+84
a) give the domain of f in interval novation
b) find the critical numbers
c) determine intervals where f is increasing and decreasing
Use the derivative test to determine
a) relative maxima
b) relative minima
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
F%28x%29+=1%2F%28x%5E2+%2B18x%2B84%29


a) give the domain of f in interval novation
domain: r all real numbers (no matter what is x, denominator will not be equal to zero
in interval novation:(-infinity,infinity)

b) find the critical numbers
critical points are points where the function is defined and its derivative is zero or undefined
so, find derivative
F'%28x%29 =%28d%2Fdx%29%281%2F%28x%5E2+%2B18x%2B84%29%29
F'%28x%29 = -%282x+%2B+18%29%2F%28x%5E2+%2B+18+x+%2B+84%29%5E2
F'%28x%29=+-%282%28x+%2B+9%29%29%2F%28x%5E2+%2B+18+x+%2B+84%29%5E2
equal it to zero
0=+-%282%28x+%2B+9%29%29%2F%28x%5E2+%2B+18+x+%2B+84%29%5E2...will be zero if numerator equal to zero
0=+-2%28x+%2B+9%29
0=x+%2B+9
x+=-9


c) determine intervals where f is increasing and decreasing
Use the derivative test to determine
if F'%28x%29%3E0 f is increasing
if F'%28x%29%3C0 f is decreasing
in your case
Increasing:(-infinity,-9)
Decreasing:(-9,infinity+

a) relative maxima
F%28x%29+=1%2F%28x%5E2+%2B18x%2B84%29, substitute x+=-9
F%28x%29+=1%2F%28%28-9%29%5E2+%2B18%28-9%29%2B84%29
F%28x%29+=1%2F%2881+-162%2B84%29
F%28x%29+=1%2F3
max at (-9,1%2F3)
b) relative minima
no global minima found

+graph%28+600%2C+600%2C+-20%2C+5%2C+-5%2C+5%2C+1%2F%28x%5E2+%2B18x%2B84%29%29+