Find a formula for the fourth degree polynomial p(x) whose graph is symmetric about the y-axis (meaning it has no odd powers of x), and which has a y-intercept of 10, and global maxima at (3,253) and (−3,253).
let the function be
y = ax^4 + bx^2 + c
since (0,10) is on it ----> c = 10
dy/dx = 4ax^3 + 2bx
at (3,253), dy/dx = 0
4a(27) + 2b(3) = 0
108a + 6b = 018a + b = 0
b = -18a
at (-3, 253) , dy/dx = 0
4a(-27) + 2b(-3) = 0 , this gives the same result, so nothing new here
but (3,253) lies on the actual curve
253 = 81a + 9b
sub in b = -18a
253 = 81a + 9(-18a)
253 = -81a
a = -253/81
then b = -18(-253/81) = 506/9
so y = (-253/81)x^4 + (506/9)x^2 + 10
check:
dy/dx = -1012/81x^3 + 1012x = 0 for a max/min
divide by -1012
x^3/81 - x = 0
times 81
x^3 - 81x = 0
x(x^2 - 9) = 0
x = 0 , or x = ±3
if x = 3 or -2
y = (-253/81)(81) + (506/9)(9)
= -253 + 506 =253 YEAHHH
(weird coefficients)
Write an equation for a fourth degree polynomial function whose graph intercepts the horizontal axis at -1/2, 0and 11.
To find the formula for a fourth-degree polynomial that meets the given criteria, we need to consider the symmetry, y-intercept, and the global maxima.
Since the graph is symmetric about the y-axis, we know that all the coefficients of odd powers of x will be zero. Therefore, we can start with the general form of a fourth-degree polynomial:
p(x) = ax^4 + bx^2 + c
Next, we can determine the values of a, b, and c using the given information.
The fact that the polynomial has a y-intercept of 10 means that when x is zero, p(x) is equal to 10. Hence:
p(0) = a(0)^4 + b(0)^2 + c = c = 10
Therefore, c = 10.
Now, we can use the fact that the polynomial has global maxima at (3,253) and (-3,253).
Substituting these values into the equation, we get:
p(3) = a(3)^4 + b(3)^2 + 10 = 253
and
p(-3) = a(-3)^4 + b(-3)^2 + 10 = 253
Expanding these equations, we have:
81a + 9b + 10 = 253
81a + 9b + 10 = 253
Simplifying further:
81a + 9b = 243
81a + 9b = 243
Dividing both sides of the equation by 9:
9a + b = 27
9a + b = 27
Since the equations are the same, the coefficients a and b must be zero.
Therefore, a = 0 and b = 0.
Substituting these values into the general form of the fourth-degree polynomial, we have:
p(x) = 0*x^4 + 0*x^2 + 10
Simplifying further, we get the final formula for the fourth-degree polynomial:
p(x) = 10
To find a formula for the fourth-degree polynomial satisfying the given conditions, we need to consider the symmetry, y-intercept, and the global maxima.
Since the graph is symmetric about the y-axis, we know that the polynomial does not contain any odd-powered terms (terms with x raised to an odd power). Therefore, we can assume that the polynomial will only contain even-powered terms.
Let's start by considering the form of the polynomial:
p(x) = ax^4 + bx^2 + c
Given that the y-intercept is at (0,10), this means that when x = 0, p(x) = 10. Plugging these values into the equation, we get:
p(0) = a(0)^4 + b(0)^2 + c = c = 10
So, we now have the value of the constant term, c.
Next, let's consider the global maxima at (3,253) and (-3,253). For these points to be maxima, the derivative of the polynomial should be equal to zero at x = 3 and x = -3.
Taking the derivative of p(x) with respect to x, we get:
p'(x) = 4ax^3 + 2bx
To find the values of a and b, we substitute x = 3 and set p'(x) = 0:
0 = 4a(3)^3 + 2b(3)
0 = 108a + 6b --------(1)
Similarly, substituting x = -3 and setting p'(x) = 0:
0 = 4a(-3)^3 + 2b(-3)
0 = -108a - 6b --------(2)
We now have a system of two equations (1) and (2) with two unknowns (a and b).
Solving this system of equations, we get:
108a + 6b = 0
-108a - 6b = 0
Adding these two equations, we get:
0 = 0
This means that the system of equations is dependent, which implies that a and b can take on any value.
Therefore, there is an infinite number of polynomials that satisfy the given conditions. A general formula for these polynomials would be:
p(x) = ax^4 + bx^2 + 10
Where a and b can be any real numbers.