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).

Answer 1

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)

Answer 2

Write an equation for a fourth degree polynomial function whose graph intercepts the horizontal axis at -1/2, 0and 11.