Find the critical numbers of the function
f(x)=x^1/6−x^−5/6.
I will guess you meant:
f(x) = x^(1/6) - x^(-5/6)
then
f'(x) = (1/6)x^(-5/6 + (5/6)x^(-11/6)
= 0 for critical values
factor it ..
(1/6)x^(-11/6)[ x + 5] = 0
x = 0 or x = - 5
sub those values back into the original function if you need the y values
To find the critical numbers of a function, we need to find the values of x for which the derivative of the function is equal to zero or undefined.
Let's find the derivative of the function f(x) = x^(1/6) - x^(-5/6) first.
Using the power rule and the chain rule, we can find the derivative of f(x):
f'(x) = (1/6) * x^(-5/6) - (-5/6) * x^(-11/6)
Simplifying the expression, we get:
f'(x) = x^(-5/6) / 6 + 5x^(-11/6) / 6
Now, let's set the derivative equal to zero and solve for x:
x^(-5/6) / 6 + 5x^(-11/6) / 6 = 0
Multiplying both sides of the equation by 6 to eliminate the denominators, we get:
x^(-5/6) + 5x^(-11/6) = 0
Now, multiply through by x^(11/6) to eliminate the negative exponents:
x^(6/6 - 5/6) + 5 = 0
x^(1/6) + 5 = 0
Subtracting 5 from both sides of the equation, we get:
x^(1/6) = -5
Now, raise both sides of the equation to the power of 6 to eliminate the fraction exponent:
(x^(1/6))^6 = (-5)^6
Simplifying:
x = 625
So, the critical number of the function f(x) = x^(1/6) - x^(-5/6) is x = 625.
To find the critical numbers of a function, we need to first take the derivative of the function and then solve for x when the derivative is equal to 0 or does not exist.
Let's start by finding the derivative of f(x). The power rule can be used for both terms in the function:
f(x) = x^(1/6) - x^(-5/6)
To find the derivative, we differentiate each term separately:
f'(x) = (1/6)x^(-5/6) + (5/6)x^(-11/6)
Now, let's set the derivative equal to 0 to find the critical numbers:
(1/6)x^(-5/6) + (5/6)x^(-11/6) = 0
To solve this equation, we need to get rid of the denominators. Multiplying both sides of the equation by 6x^(11/6), we get:
x^(11/6) + 5x^(5/6) = 0
This equation does not factor nicely, so we may need to use numerical methods or approximation techniques to find the solutions. One common method is to use a graphing calculator or software to graph the function and find the x-intercepts, which represent the critical numbers.
In this case, it is challenging to provide an exact solution without the use of numerical methods, but the critical numbers can be found by plotting the graph of the function or using numerical approximation techniques.