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Analyzing the Function f(x) = (x^4)/4 + (2/3)x^3 - (5/2)x^2 - 6x + 7
This article will delve into the analysis of the function f(x) = (x^4)/4 + (2/3)x^3 - (5/2)x^2 - 6x + 7. We will explore its key characteristics, including its domain, range, intercepts, symmetry, and end behavior.
Domain and Range
The function f(x) is a polynomial function, meaning it's defined for all real numbers. Therefore, the domain of f(x) is (-∞, ∞).
Determining the range of a polynomial function requires further analysis. We can start by examining the leading term (x^4/4) and its coefficient (1/4). Since the leading coefficient is positive and the degree of the polynomial is even, the function will have positive infinity as its end behavior on both sides. This suggests that the range of f(x) is also (-∞, ∞).
Intercepts
x-intercepts are the points where the graph of the function crosses the x-axis. To find them, we set f(x) = 0 and solve for x.
(x^4)/4 + (2/3)x^3 - (5/2)x^2 - 6x + 7 = 0
Solving this equation for x can be challenging and may require numerical methods or graphing tools.
y-intercepts are the points where the graph crosses the y-axis. To find them, we set x = 0 and evaluate f(x):
f(0) = (0^4)/4 + (2/3)(0^3) - (5/2)(0^2) - 6(0) + 7 = 7
Therefore, the y-intercept is (0, 7).
Symmetry
To determine the symmetry of f(x), we can check if it satisfies the following conditions:
- Even Function: f(-x) = f(x)
- Odd Function: f(-x) = -f(x)
Let's check if f(x) is even or odd:
f(-x) = (-x)^4/4 + (2/3)(-x)^3 - (5/2)(-x)^2 - 6(-x) + 7 f(-x) = (x^4)/4 - (2/3)x^3 - (5/2)x^2 + 6x + 7
As f(-x) is not equal to f(x) or -f(x), the function is neither even nor odd. This indicates that the graph of f(x) does not possess any symmetry about the y-axis or the origin.
End Behavior
As mentioned earlier, the leading term (x^4/4) dictates the end behavior of the function. Because the coefficient of the leading term is positive and the degree is even, the graph of f(x) rises to positive infinity as x approaches both positive and negative infinity.
In summary, the end behavior of f(x) is:
- x → ∞, f(x) → ∞
- x → -∞, f(x) → ∞
Further Analysis
To obtain a more comprehensive understanding of f(x), we can use calculus techniques. By finding the derivative, f'(x), we can determine the critical points, intervals of increase and decrease, and local extrema. The second derivative, f''(x), will reveal the concavity and inflection points of the function.
By applying these methods, we can construct a detailed graph of the function f(x) and gain insights into its behavior.