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)=
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
To find a formula for the fourth-degree polynomial that satisfies the given conditions, we can follow these steps:
Step 1: Symmetry about the y-axis
Since the graph is symmetric about the y-axis, we know that the polynomial will have even powers only (i.e., x^0, x^2, x^4).
Step 2: Y-intercept
The y-intercept is given as 4. This means that when x = 0, p(x) = 4. Since the polynomial is symmetric about the y-axis, the coefficient of x^0 (constant term) will be 4.
Step 3: Global maxima
The global maxima are given at (1,7) and (-1,7). This means that the polynomial touches the x-axis at x = 1 and x = -1. Thus, the two factors in the polynomial will be (x - 1) and (x + 1).
Step 4: Assemble the polynomial
We now have the following components: y-intercept (constant term = 4) and the two linear factors ((x - 1) and (x + 1)). To create a fourth-degree polynomial, we need two more terms with even powers. Let's call these terms a and b. The final formula for the polynomial becomes:
p(x) = a(x - 1)(x + 1) + b(x - 1)^2(x + 1)^2
Step 5: Use global maxima to find a and b
Using the global maxima, we can determine the values of a and b. When x = 1, p(x) = 7, and when x = -1, p(x) = 7. Substituting these values into the polynomial equation, we get two equations:
7 = a(1 - 1)(1 + 1) + b(1 - 1)^2(1 + 1)^2 ---> (Equation 1)
7 = a(-1 - 1)(-1 + 1) + b(-1 - 1)^2(-1 + 1)^2 ---> (Equation 2)
Simplifying these equations will give us the values for a and b.
Step 6: Substitute the values of a and b into the polynomial
Once we have the values of a and b, we substitute them back into the polynomial equation:
p(x) = a(x - 1)(x + 1) + b(x - 1)^2(x + 1)^2
This will give us the final formula for the fourth-degree polynomial that satisfies the given conditions.