Let f(x)= -3x^4+79x^2-3x+1
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5x^4+19x^4+2x+5
Discuss the short run behavior for f(x)and the long run for f(x).
did you not check on previous replies to your previous questions?
see:
http://www.jiskha.com/display.cgi?id=1197011555
I think you meant to write 19x^3, not 19x^4, in the denominator
As x-> +infinity or -infinity, f(x) approaches an asymptote of -3/5
At x=0, it equals 1/5. In the region "near" x=0, there are some wild oscillations, and probably a few crossings of the x axis (where the numerator equals zero). There may be some singularities (infinities)in a negative-x region were the denominator is zero.
I suggest you do some plotting of the function yourself.
To discuss the short run and long run behavior of the function f(x) = (-3x^4 + 79x^2 - 3x + 1) / (5x^4 + 19x^4 + 2x + 5), we need to understand the behavior of the function as x approaches certain values.
1. Short run behavior: The short run behavior refers to the behavior of the function around specific x-values, typically near points of interest such as critical points, local extrema, or points of discontinuity.
To determine the short run behavior of f(x), we need to find the critical points of the function. Critical points occur where the derivative of the function is either zero or undefined.
To find the critical points:
Step 1: Differentiate f(x) with respect to x.
Step 2: Set the derivative equal to zero and solve for x.
Step 3: Determine the x-values for which the derivative is undefined (if any).
After finding the critical points, we can analyze the behavior of f(x) around these points by evaluating the function at values slightly greater and slightly smaller than the critical points.
2. Long run behavior: The long run behavior of a function describes its behavior as x approaches positive or negative infinity. It helps us understand the overall trend of the function for very large or very small x-values.
To determine the long run behavior of f(x), we examine the leading terms in the numerator and denominator. The leading term is the one with the highest exponent.
In this case, the leading term in the numerator is -3x^4, and the leading term in the denominator is 24x^4. As x approaches positive or negative infinity, the higher-order terms dominate, and the function behaves similarly to the ratio of their coefficients.
So, in the long run, for very large or very small x-values, f(x) behaves similar to the ratio of -3x^4/24x^4 = -1/8, which means the function approaches the constant value -1/8.
To summarize:
- The short run behavior of f(x) involves analyzing the function near specific x-values (critical points, local extrema, or points of discontinuity).
- The long run behavior of f(x) describes the overall trend of the function as x approaches positive or negative infinity and is determined by the leading terms in the numerator and denominator. In this case, f(x) approaches -1/8 in the long run.