Find the values of a , b , and c in the quadratic function p(x)=ax2+bx+c such that p(2)=6, p'(2)=2, p"(2)=3
p(2) = 4a + 2b + c = 6
p'(x) = 2ax + b
p'(2) = 4a + b = 2 ---> b = 2 - 4a
b = 1 - 2a
p''(x) = 2a = 3
a = 3/2
b = 2 - 4(3/2) = -4
in 1st equation:
4(3/2) - 8 + c = 6
c = 8
To find the values of a, b, and c in the quadratic function p(x) = ax^2 + bx + c, we need to use the given information about p(2), p'(2), and p"(2).
Step 1: Finding p(2)
We are given that p(2) = 6. To find this, substitute x = 2 into the quadratic function:
p(2) = a(2)^2 + b(2) + c
6 = 4a + 2b + c
Step 2: Finding p'(2)
We are given that p'(2) = 2. To find this, we need to find the derivative of the quadratic function and evaluate it at x = 2.
p'(x) = 2ax + b
p'(2) = 2a(2) + b
2 = 4a + b
Step 3: Finding p"(2)
We are given that p"(2) = 3. To find this, we need to find the second derivative of the quadratic function and evaluate it at x = 2.
p"(x) = 2a
p"(2) = 2a
3 = 2a
Now, we have a system of three equations with three unknowns (a, b, c):
1) 6 = 4a + 2b + c
2) 2 = 4a + b
3) 3 = 2a
We can solve this system of equations to find the values of a, b, and c.
Let's solve this system using the substitution method:
From equation 3, we have a = 3/2. Substitute this value in equation 2:
2 = 4(3/2) + b
2 = 6 + b
b = 2 - 6
b = -4
Substitute the values of a = 3/2 and b = -4 in equation 1:
6 = 4(3/2) + 2(-4) + c
6 = 6 - 8 + c
6 = -2 + c
c = 6 + 2
c = 8
Therefore, the values of a, b, and c are:
a = 3/2
b = -4
c = 8
So, the quadratic function p(x) = (3/2)x^2 - 4x + 8 satisfies the given conditions p(2) = 6, p'(2) = 2, and p"(2) = 3.