Consider the function f(x)=2x+9x^–1.?
For this function there are four important intervals: (-inf,–A], [A,B),(B,C), and [C,inf)
where A, and C are the critical numbers and the function is not defined at B.
Find A= ?????
and B= 0
and C =?????
i have this one already
In that case I will assume that you no longer need help.
Since f(x) = 2x + 9/x
f'(x) = 2 - 9/x^2
The function has extreme values at
x = 3/sqrt2 and = -3/sqrt2
Those are the points A and C.
To find the critical numbers of the function f(x) = 2x + 9x^(-1), we need to determine where the derivative of the function is equal to zero or undefined.
First, let's find the derivative of f(x). The derivative of 2x is simply 2, and the derivative of 9x^(-1) can be found using the power rule. The power rule states that the derivative of x^(-n) is equal to -n*x^(-n-1). Applying this rule, we get -9x^(-2) as the derivative of 9x^(-1).
So, the derivative of f(x) is f'(x) = 2 - 9x^(-2).
Now, we need to set f'(x) equal to zero and solve for x to find the critical numbers.
2 - 9x^(-2) = 0
We can rearrange this equation as follows:
2 = 9x^(-2)
Dividing both sides by 9, we have:
2/9 = x^(-2)
To eliminate the negative exponent, we can take the reciprocal of both sides of the equation:
9/2 = x^2
Taking the square root of both sides, we get:
√(9/2) = x
Simplifying the square root, we have:
√(9/2) = ±3/√2 = ±3√2/2
Thus, the critical numbers are A = -3√2/2 and C = 3√2/2.
The function is not defined at B. In this case, B = 0.
To find the critical numbers of the function f(x) = 2x + 9x^(-1), we need to find the values of x that make the derivative of the function equal to zero or undefined.
To find A, we need to find the values that make the denominator of the function equal to zero, which would result in an undefined value. In this case, the denominator is 9x^(-1), which is equivalent to 9/x. Setting the denominator equal to zero:
9/x = 0
Multiplying both sides by x:
9 = 0
This equation has no solution because you cannot divide any number by zero. Therefore, there is no critical number at A.
For B, you mentioned that the function is not defined at B. Based on the information given, B = 0.
To find C, we need to find the values that make the derivative of the function equal to zero. The derivative of f(x) = 2x + 9x^(-1) can be found using the power rule and the quotient rule:
f'(x) = 2 - 9/x^2
Setting the derivative equal to zero:
2 - 9/x^2 = 0
Rearranging the equation:
2 = 9/x^2
Multiplying both sides by x^2:
2x^2 = 9
Dividing both sides by 2:
x^2 = 9/2
Taking the square root of both sides:
x = ±√(9/2)
So, C = ±√(9/2).
Therefore, the critical numbers for the function f(x) = 2x + 9x^(-1) are A = N/A (no critical number), B = 0, and C = ±√(9/2).