F(x)=square root(27x-x^4)
To find the expression of f(x), which is given by f(x) = √(27x - x^4), we start by analyzing the components of the function.
The function f(x) consists of two main parts: the square root (√) and the expression inside the square root (27x - x^4).
Let's break it down step by step:
1. Start with the expression 27x - x^4:
- This is a difference of two terms: 27x and x^4.
- This means you have a fourth-degree polynomial expression, where x is raised to the 4th power.
2. Take the square root of 27x - x^4:
- Taking the square root of an expression involves finding the value that, when squared, equals the given expression.
- In this case, we take the square root of 27x - x^4, which means we need to find a value for x that makes 27x - x^4 non-negative.
Note: For simplicity, let's assume that x is a real number.
3. Determine the conditions:
- The expression 27x - x^4 must be greater than or equal to zero, since the square root of a negative number is undefined.
4. Solve the conditions by finding the critical points:
- To solve the inequality 27x - x^4 ≥ 0, we need to find the points where the expression changes its sign or value.
- We do this by finding the critical points, which are the points where the expression equals zero or is undefined.
To find the critical points:
- Set 27x - x^4 = 0, and solve for x.
- Factor out an 'x' from the equation: x(27 - x^3) = 0.
- The critical points occur when x = 0, or when x^3 = 27 (which implies x = 3, as 3^3 = 27).
5. Determine the intervals:
- Now that we have the critical points (x = 0 and x = 3), we can divide the number line into intervals:
- Interval 1: x < 0
- Interval 2: 0 < x < 3
- Interval 3: x > 3
6. Analyze the intervals with the conditions:
- We need to check if the expression 27x - x^4 is non-negative or zero in each interval.
- Plug in a test value within each interval into the expression to determine its sign.
For Interval 1 (x < 0):
- Choose x = -1 (as a test value): 27(-1) - (-1)^4 = -27 - 1 = -28.
- Since -28 is negative, the expression 27x - x^4 < 0 for any x in this interval.
For Interval 2 (0 < x < 3):
- Choose x = 1 (as a test value): 27(1) - (1)^4 = 27 - 1 = 26.
- Since 26 is positive, the expression 27x - x^4 > 0 for any x in this interval.
For Interval 3 (x > 3):
- Choose x = 4 (as a test value): 27(4) - (4)^4 = 108 - 256 = -148.
- Since -148 is negative, the expression 27x - x^4 < 0 for any x in this interval.
7. Combine the intervals and the corresponding signs:
- The expression 27x - x^4 is positive (or zero) for 0 < x < 3.
- Therefore, we can take the square root of 27x - x^4 in this interval.
8. Write the expression f(x):
- Combining the square root (√) and the expression 27x - x^4, we have:
f(x) = √(27x - x^4) for 0 < x < 3.
So, that's how you find and analyze the expression f(x) = √(27x - x^4).