final question
Consider the function F(x)=3x^4-5x+3 / X^4+1
Find:
Domain
Vertical,horizontal,or slant asymptotes
x-intercept(s)
y-itercept(s)
symmetry(respect to x, y-axis or orgin)
F’ (x)
Critical numbers
F” (x)
Possible points of inflection
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This problem is a good pre-exam example on functions, since it covers most of the aspects.
f(x)=(3x^4-5x+3)/(x^4+1)
DOMAIN
dom f(x)=dom(numerator)∪dom(denominator) - undefined points
Since both numerator and denominator are polynomials, dom is (-∞,∞). The denominator is positive for all real values of x, so there are no undefined (singular) points.
Therefore dom f(x) = (-∞,∞).
ASYMPTOTES
The function is continuous in its domain and the denominator is always positive, so there are no vertical asymptotes.
To find the horizontal asymptotes, if any, we take limits of the function as x->-∞(=3) and as x->+∞(=3). So there is a horizontal asymptote at y=3. To find from which side of y=3 the function approches the asymptote, we consider the term after the leading term in the numerator, -5x. Since -5x>0 when x->-∞ and -5x<0 when x->∞, we conclude that the function approaches the horizontal asymptote from 3+ as x->-∞ and from 3- as x->+∞.
INTERCEPTS
x-intercepts, if any, will be the real roots of f(x)=0.
Since the function cannot be factorized, it is not obvious if there are x-intercepts. This can be confirmed after finding the local extrema following the calculation of critical points.
y-intercept(s)
Since this is a function, there cannot be more than one y-intercept (ref. vertical line test).
The y-intercept equals f(0)=3.
Symmetries
f(x)≠f(-x), so it is not an even function, i.e. it is not symmetric about the y-axis.
f(x)≠-f(-x), so f(x) is not an odd function, i.e. it is not symmetrical with respect to the origin.
f(x) cannot be symmetrical about the x-axis, or else it will fail the vertical line test (i.e. not a function).
FIRST DERIVATIVE OF f(x)
The first derivative of f(x) can be calculated by the quotient rule and carefully simplifying the resulting expression.
f'(x) = 5(3x^4-1)/(x^4+1)²
If we factorize the numerator, we can easily find the zeroes of f'(x)=0:
5(3x^4-1)
=15(x²-√3/3)(x²+√3/3)
=15(x+3-1/4)(x-3-1/4)(x²+√3/3)
from which the zeroes of f'(x) can be readily deduced as
x=±3-1/4=±0.76.
We will denote these two values of x as
c1=-3-1/4 and
c2= 3-1/4
CRITICAL NUMBERS (or critical points) and END POINTS
If the function is defined on the interval [a,b], the points x=a and x=b are the end-points of the function. These are also points that are candidates of local or global extrema.
For the given function, the domain being (-∞,&infin), there are no end-points.
Critical points correspond to values of x for which f'(x)=0 or where f'(x) is undefined. Together with end-points, these are also the only points where local and global extrema can occur.
The only critical points for this function occur at x=c1 and x=c2. Since the horizontal asymptote is at y=3, the only places for a global maximum and global minimum can occur are at x=c1 and x=c2.
from f(c1)=5.84 (approx.) and f(c2)=0.15 (approx.), we conclude that the range of f(x) is [f(c2),f(c1)].
Since f(c1)>0 and f(c2)>0, and the horizontal asymptote y=3 >0, we conclude that f(x) has no real zeroes. SO THERE ARE NO X-INTERCEPTS.
Apply the first derivative test to confirm that (c1,f(c1)) is a local maximum, and (c2,f(c2)) is a local minimum.
Together with the horizontal asymptote, we conclude also that (c1,f(c1)) is a global maximum, and (c2,f(c2)) is a global minimum, since there are no end-points or other critical points.
SECOND DERIVATIVE OF f(x)
The second derivative is useful in
1. confirming the nature of the local extrema (local max. or minimum)
2. determining the concavity of the graph of the function.
3. determining the inflection points.
First, we need to find the second derivative of f(x).
This can be done again using the quotient rule and carefully simplifying the resulting expression, which becomes:
f"(x)=-(20*x³*(3*x⁴-5))/(x^4+1)³
f"(c1)=-14.8 <0, therefore (c1,f(c1)) is a maximum.
f"(c2)=+14.8 >0 therefore (c2,f(c2)) is a minimum.
The above conclusions confirm the findings of the first derivative test.
Next, we determine the zero(es) of f"(x) which are readily found to be x=0, x=±(5/3)1/4 (=±1.14 approx.).
These will be denoted by
i1=-(5/3)1/4
i2=0
i3=(5/3)1/4
and they represent the points of inflection.
Since there are three inflection points, there are three changes of concavity.
For (-∞,i1], we evaluate f"(-2)=+1.4, therefore the graph is concave up. This also confirms the earlier observation that the graph approaches the horizontal asymptote from 3+.
For [i1,0], we evaluate f"(-1)=-5, meaning that the graph is concave down. Continuing similarly, we find [0,i2] is concave up, and [i2,∞] is concave down, again confirming that the graph approaches the horizontal asymptote from y=3-.
The graph of the function can be found at the following links:
http://img16.imageshack.us/img16/4287/1260035879a.png
http://img31.imageshack.us/img31/1139/1260035879b.png
To find the domain of the function F(x), we need to determine any restrictions on the values of x. In this case, there is no restriction or division by zero in the function. Therefore, the domain of F(x) is all real numbers.
To find the vertical, horizontal, or slant asymptotes, we need to analyze the behavior of the function at the extremes. Let's consider the limit as x approaches positive or negative infinity:
lim(x→∞) F(x) = lim(x→∞) (3x^4-5x+3) / (x^4+1) = ∞/∞ (indeterminate form)
In this case, we can apply L'Hopital's Rule to find the limit of the ratio of the derivatives of the numerator and denominator. Differentiating the numerator and denominator:
lim(x→∞) F(x) = lim(x→∞) (12x^3-5) / (4x^3) = lim(x→∞) (12 - 5/x^3) / 4 = 12/4 = 3
Therefore, as x approaches positive or negative infinity, the function approaches a horizontal asymptote at y = 3.
To find the x-intercepts, we set F(x) equal to zero and solve for x:
3x^4 - 5x + 3 = 0
Unfortunately, this is a quartic equation, which is generally difficult to solve analytically. You can use numerical methods or approximation techniques to find the x-intercepts.
To find the y-intercept, we set x equal to 0 and evaluate F(x):
F(0) = (3(0)^4-5(0)+3) / ((0)^4+1) = 3/1 = 3
Therefore, the y-intercept is (0, 3).
To determine the symmetry of the function, we can substitute x with -x and see if the function remains the same:
F(-x) = (3(-x)^4-5(-x)+3) / ((-x)^4+1) = (3x^4 + 5x + 3) / (x^4 + 1)
Since F(-x) is not equal to F(x), the function is not symmetric with respect to the y-axis or the origin.
To find F'(x), we need to differentiate the function with respect to x:
F'(x) = [(12x^3)(x^4+1) - (3x^4 - 5x + 3)(4x^3)] / (x^4+1)^2
To find the critical numbers, we set F'(x) equal to zero and solve for x:
[(12x^3)(x^4+1) - (3x^4 - 5x + 3)(4x^3)] / (x^4+1)^2 = 0
Solving this equation could be quite challenging analytically as well. You may need to use numerical methods to find the critical numbers.
To find F''(x), we differentiate F'(x):
F''(x) = [(216x^2)(x^4+1)^2 - 2(x^4+1)(12x^3)(4x^3)] / (x^4+1)^4
To find possible points of inflection, we set F''(x) equal to zero:
[(216x^2)(x^4+1)^2 - 2(x^4+1)(12x^3)(4x^3)] / (x^4+1)^4 = 0
Again, solving this equation analytically might be difficult. Numerical methods may be required to find the points of inflection.
To find the domain of a function, we need to determine the values of x for which the function is defined. In this case, the denominator of the function, x^4 + 1, cannot be zero because division by zero is undefined. Since x^4 + 1 is always positive for any real value of x, there are no restrictions on the domain. Therefore, the domain of the given function is all real numbers.
To find the vertical asymptotes, we need to check for values of x for which the denominator of the function becomes zero. However, in this case, x^4 + 1 can never equal zero because x^4 is always non-negative and adding 1 to it will always give a positive value. Therefore, there are no vertical asymptotes.
To find the horizontal asymptotes, we need to examine the behavior of the function as x approaches positive or negative infinity. Since the degree of the numerator is equal to the degree of the denominator, we perform the division of the leading coefficients. In this case, the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Dividing 3 by 1 gives us 3. Therefore, the horizontal asymptote is y = 3.
To find the x-intercepts, we set the numerator of the function (3x^4 - 5x + 3) equal to zero and solve for x. However, this requires solving a quartic equation, which can be quite involved. It may be difficult to find the exact values of the x-intercepts algebraically. Therefore, you may need to use numerical methods or graphing technology to find the approximate values.
To find the y-intercept, we can plug in x = 0 into the function and evaluate it. Substituting x = 0 into the function F(x), we have F(0) = (3(0)^4 - 5(0) + 3) / (0^4 + 1) = 3/1 = 3. Therefore, the y-intercept is (0, 3).
To determine symmetry with respect to the x-axis, we check if the function remains unchanged when all y-values are replaced by their negations. In this case, the function is not symmetric with respect to the x-axis because changing the sign of y does not result in an identical function.
To determine symmetry with respect to the y-axis, we check if the function remains unchanged when all x-values are replaced by their negations. In this case, the function is not symmetric with respect to the y-axis because changing the sign of x does not result in an identical function.
To determine symmetry with respect to the origin, we check if the function remains unchanged when both x-values and y-values are replaced by their negations. In this case, the function is not symmetric with respect to the origin because changing the sign of both x and y does not result in an identical function.
To find the derivative F'(x) of the function, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative f'(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Using the quotient rule on the given function F(x), where g(x) = 3x^4 - 5x + 3 and h(x) = x^4 + 1, we can differentiate g(x) and h(x) separately to obtain their respective derivatives. Then we substitute these derivatives into the formula and simplify to find F'(x).
To find the critical numbers, we set the derivative F'(x) equal to zero and solve for x. The critical numbers are the values of x where the derivative is either zero or undefined. However, in this case, it may be quite complicated to find the exact values of the critical numbers algebraically due to the high degree of the polynomial functions involved. It may be more practical to use numerical methods or graphing technology to approximate the critical numbers.
To find the second derivative F''(x) of the function, we differentiate the derivative F'(x) obtained earlier using the quotient rule. We repeat the process of differentiating g'(x) and h'(x) separately, and then substitute these derivatives into the formula for F''(x):
F''(x) = [(g''(x) * h(x) - g(x) * h''(x)) * (h(x))^2 - (g'(x) * h(x) - g(x) * h'(x)) * 2 * h(x) * h'(x)] / (h(x))^4
Once again, finding the second derivative may involve lengthy algebraic calculations due to the high degree of the polynomial functions. Numerical methods or graphing technology can be helpful in approximating the second derivative and its corresponding points.
To find the possible points of inflection, we need to find the x-values where the concavity of the function changes. This occurs when the second derivative F''(x) equals zero or is undefined. However, exact values for these points may be difficult to determine algebraically. Numerical methods or graphing technology can assist in finding the approximate x-values of the inflection points.