If it has 2 global maxima, then it has a local minimum at x=0 (due to the symmetry). So,
p' = ax(x^2-1) = a(x^3-x)
That means
p = a (1/4 x^4 - 1/2 x^2) + c
We know that p(0) = 4.So c = 4 and
p(x) = a (1/4 x^4 - 1/2 x^2) + 4
Since p(1) = 7,
a(1/4 - 1/2) + 4 = 7
-1/4 a = 3
a = -12
p(x) = -3x^4 + 6x^2 + 4
see the graph at
https://www.wolframalpha.com/input/?i=+-3x%5E4+%2B+6x%5E2+%2B+4
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 (1,7) and (−1,7).
p(x)=
1 answer