(1 point)
Find the positive critical point of the function f(x)=x/x^6+9
To find the positive critical point of the function f(x) = x/(x^6 + 9), we need to find the derivative of the function and set it equal to zero.
Step 1: Find the derivative of f(x):
To find the derivative of f(x), we will use the quotient rule. The quotient rule states that the derivative of (u/v) is equal to (v * du/dx - u * dv/dx) / (v^2).
In this case, u = x and v = x^6 + 9. Therefore, du/dx = 1 and dv/dx = 6x^5.
Using the quotient rule, we can calculate the derivative of f(x):
f'(x) = (x^6 + 9)(1) - (x)(6x^5) / (x^6 + 9)^2
Simplifying the expression:
f'(x) = (x^6 + 9 - 6x^6) / (x^6 + 9)^2
= (9 - 5x^6) / (x^6 + 9)^2
Step 2: Set f'(x) equal to zero:
To find the critical points, we set the derivative f'(x) equal to zero and solve the equation:
(9 - 5x^6) / (x^6 + 9)^2 = 0
Since we are looking for positive critical points, we can ignore the denominator (x^6 + 9)^2, as it is always positive. Our equation becomes:
9 - 5x^6 = 0
Step 3: Solve for x:
Subtract 9 from both sides:
-5x^6 = -9
Divide both sides by -5:
x^6 = 9/5
Take the sixth root of both sides:
x = (9/5)^(1/6)
This is the positive critical point of the function f(x) = x/(x^6 + 9).