Calculus
posted by Dana .
Find a formula for the fourth degree polynomial p(x) whose graph is symmetric about the yaxis (meaning it has no odd powers of x), and which has a yintercept 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)