Question

Use the first derivative to determine where the function
f(x)=4x-9x^(1/2) is increasing and decreasing

Answer 1

can you take the derivative?

Answer 2

4-(9/2)x^(-1/2)

Answer 3

ok, good. so whenever that function is negative, it means the original function is decreasing and whenever its positive it means the original function is increasing.

Answer 4

so for what values of x is your derivative positive/negative

Answer 5

iam confused

Answer 6

if your derivative 4-(9/2)x^(-1/2) is negative it means you function f(x)=4x-9x^(1/2) has a negative slope at that point so its decreasing. If the derivative is positive, the function f(x) has a positive slope and is increasing

Answer 7

so what do we do next

Answer 8

you dreivative is f'(x)=4-(9/2)x^(-1/2) or f'(x) = 4 - (9/2)* (1/x^.5)
so you'll find that x^.5 is the same as sqrt(x) right? and you cant have a negative under the square root so all x values must be positive. right?

Answer 9

yes

Answer 10

good so if the term (9/2)*(1/x^.5)) > 4 the function f(x) will be decreasing because the derivative will be negative at that point, if its (9/2)*(1/x^.5)) < 4, it f(x) will be increasing because the derivative is positive

Answer 11

the point where (9/2)*(1/x^.5)) = 4 is when f'(x)=0 and this is the point where if(x) switches from being a decreasing function to an increasing function

Answer 12

hope that helps