Find the positive critical point of the function f(x)=x/(x^6+9)
f = x/(x^6+9)
f' = (9-5x^6)/(x^6+9)^2
f" = 6x^5(5x^6-53)/(x^6+9)^3
since the denominator is never zero, we just need to find where
6x^5(5x^6-53) = 0
That is at x=0, x = ±(53/6)^(1/6) = ±1.4377
To find the positive critical point of the given function f(x) = x/(x^6+9), we need to find the value of x where the derivative of the function is equal to zero.
Step 1: Find the derivative of f(x) with respect to x using the quotient rule.
f'(x) = [x'*(x^6+9) - x*(6x^5)] / (x^6+9)^2
= (x^6+9 - 6x^6) / (x^6+9)^2
= (9 - 5x^6) / (x^6+9)^2
Step 2: Set f'(x) equal to zero and solve for x.
(9 - 5x^6) / (x^6+9)^2 = 0
To make a fraction equal to zero, the numerator must be equal to zero.
9 - 5x^6 = 0
Step 3: Solve the equation for x.
5x^6 = 9
x^6 = 9/5
x = (9/5)^(1/6)
The positive critical point of the given function f(x) = x/(x^6+9) is x = (9/5)^(1/6).