Find the values of a , b , and c in the quadratic function p(x)=ax^2+bx+c such that p(2)=7, p′(2)=4, and p′′(2)=3.
p' = 2ax + b
p" = 2a
so, 2a = 3
a = 3/2
p' = 3x+b
p'(2) = 4, so 12+b = 4; b = -8
p = 3/2 x^2 - 8x + c
p(2) = 7, so 6-16+c = 4; c = 16
p(x) = 3/2 x^2 - 8x + 16
oops. catch my typo (in p'(2))
To find the values of a, b, and c in the quadratic function p(x) = ax^2 + bx + c, we will use the given information.
Step 1: Using p(2) = 7
Substitute x = 2 into the equation and solve for a, b, and c:
p(2) = a(2^2) + b(2) + c = 7
4a + 2b + c = 7 -- Equation 1
Step 2: Using p'(2) = 4
Differentiate p(x) with respect to x:
p'(x) = 2ax + b
Substitute x = 2 into the derivative equation:
p'(2) = 2a(2) + b = 4
4a + b = 4 -- Equation 2
Step 3: Using p''(2) = 3
Differentiate p'(x) with respect to x:
p''(x) = 2a
Substitute x = 2 into the second derivative equation:
p''(2) = 2a = 3
a = 3/2 -- Equation 3
Step 4: Solve for b and c
Substitute the value of a from Equation 3 into Equation 2:
4(3/2) + b = 4
6 + b = 4
b = 4 - 6 = -2
Substitute the values of a = 3/2 and b = -2 into Equation 1:
4(3/2) + 2(-2) + c = 7
6 - 4 + c = 7
c - 2 = 7
c = 7 + 2 = 9
Therefore, the values of a, b, and c are a = 3/2, b = -2, and c = 9 in the quadratic function p(x) = (3/2)x^2 - 2x + 9.