Find a second degree polynomial p such that p(2) = 5, p'(2) = 3, and p''(2) = 2

Solution:

Given polynomial p such that p(2) = 5, p'(2) = 3, and p''(2) = 2

Let the polynomial be ax² + bx + c = 0 = p

Put x = 2, a(2)² + b(2) + c = 5

4a + 2b + c =5 --- [1]

P’ = 2ax + b

p’(2) = 2a(2) + b = 3

4a + b = 3 --- [2]

p’’ = 2a = 2

a = 1

Put a = 1 in eq[2], we get

4(1) + b = 3

b = 3 - 4

b = -1

Put a = 1 and b = -1 in eq[1],

we get 4(1) + 2(-1) + c = 5

4 - 2 + c = 5

2 + c = 5

c = 5-2

c = 3

Therefore, the polynomial is x² - x + 3 = 0

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Find a second degree polynomial p such that p(2) = 5, p'(2) = 3, and p''(2) = 2

Summary:

A second degree polynomial p such that p(2) = 5, p'(2) = 3, and p''(2) = 2 is x² - x + 3 = 0.