Consider the function
f(x) = 7 x + 3 {x ^ -1 }. For this function there are four important intervals: (-infinity, A], [A,B),(B,C), and [C,infinity) where A, B and C are either critical numbers or points at which f(x) is undefined. Find A, B and C
To find the critical numbers or points at which the function is undefined, you need to consider the different parts of the function separately.
1. Start by identifying the points where the function becomes undefined due to division by zero. In this case, the function is undefined when the denominator of the term {x ^ -1 } equals zero. Rearranging the equation x ^ -1 = 0, we get x = 0. Therefore, x = 0 is a point where the function is undefined.
2. Next, find the critical numbers by taking the derivative of the function. For the given function f(x) = 7x + 3 {x ^ -1 }, we can split it into two terms: 7x and 3 {x ^ -1 }. The derivative of 7x is simply 7, while the derivative of 3 {x ^ -1 } can be found using the power rule. Taking the derivative, we get:
f'(x) = 7 - 3 {x ^ -2 }
To find the critical numbers, set the derivative equal to zero and solve for x:
7 - 3 {x ^ -2 } = 0
3 {x ^ -2 } = 7
{x ^ -2 } = 7/3
{x ^ -2 } = (3 * 7) / 3
{x ^ -2 } = 21/3
{x ^ -2 } = 7
Taking the reciprocal of both sides, we get:
x ^ 2 = 1/7
Taking the square root, we find the positive and negative values for x:
x = ±√(1/7)
So, ±√(1/7) are the critical numbers.
3. Now we have all the necessary information to determine the values of A, B, and C:
- A corresponds to negative infinity since it's the lower limit of the first interval (-∞, A]. So, A = -∞.
- B corresponds to the value between the critical numbers -√(1/7) and √(1/7), because the interval is indicated as (A, B). So, B = √(1/7).
- C corresponds to positive infinity since it's the upper limit of the interval [C, ∞). So, C = ∞.
In summary, the values of A, B, and C are:
A = -∞
B = √(1/7)
C = ∞