Find a formula for the fourth degree polynomial p(x) whose graph is symmetric about the y-axis, and which has a y-intercept of 4, and global maxima at (1,6) and(-1,6).
y = a x^4 + b x^2 + c if symmetric about x = 0
when x = 0, y = 4
so
c = 4 and
y = a x^4 + b x^2 + 4
when x = 1, y = 6
6 = a + b + 4
so
a+b = 2
b = (2-a)
dy/dx = 4 a x^3 + 2 b x
= 0 when x = 1 or - 1
0 = 4 a + 2 b
0 = 4 a + 2(2-a)
0 = 4 a + 4 -2a
2 a = -4
a = - 2
b = 4
y = -2 x^4 + 4 x^2 + 4
the symmetry tells us that there is a local minimum at y=0
So, p'(x) = ax(x-1)(x+1) = a(x^3-x)
p(x) = a/4 x^4 - a/2 x^2 + c
since p(0) = 4,
p(x) = a/4 x^4 - a/2 x^2 + 4
Now, we know p(1)=p(-1) = 6, so
6 = a/4 - a/2 + 4
a = -8
so,
p(x) = -2x^4 + 4x^2 + 4
To find a formula for the fourth degree polynomial that satisfies the given conditions, we can start by considering the symmetry of the graph. Since the graph is symmetric about the y-axis, the polynomial will have only even degree terms and no odd degree terms.
Let's assume the general form of the polynomial is:
p(x) = ax⁴ + bx² + c
Since the graph has a y-intercept of 4, we know that when x = 0, p(x) = 4. Substituting these values:
p(0) = a(0)⁴ + b(0)² + c = 4
c = 4
So now our polynomial becomes:
p(x) = ax⁴ + bx² + 4
Next, we need to use the given global maxima at (1,6) and (-1,6) to determine the values of a and b.
First, let's consider (1,6). We can substitute the values in the polynomial:
p(1) = a(1)⁴ + b(1)² + 4 = 6
a + b + 4 = 6
Next, let's consider (-1,6):
p(-1) = a(-1)⁴ + b(-1)² + 4 = 6
a + b + 4 = 6
Since both equations yield the same result, we can write:
a + b + 4 = 6
Now, solve for a + b:
a + b = 2
We have two unknowns (a and b) and only one equation, so we need one more equation to solve for a and b. We can use the condition that the graph is symmetric about the y-axis.
For the graph to be symmetric about the y-axis, the coefficients of the odd degree terms must be zero. In this case, it means that the coefficient of x² should be zero.
So we have:
b = 0
Now we can substitute this into our equation a + b = 2:
a + 0 = 2
a = 2
Finally, we can substitute the values of a, b, and c back into the polynomial:
p(x) = 2x⁴ + 4
Therefore, the formula for the fourth degree polynomial p(x) that satisfies the given conditions is:
p(x) = 2x⁴ + 4