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