Numbers!

Calculus Level 2

The numbers p , q , r p, q, r such that the quadratic function f ( x ) = p x 2 + q x + r f(x) = px^2 + qx + r satisfies the condition f ′ ( 1 ) = 4 , f ( 2 ) + f ′ ′ ( 2 ) = 13 f'(1) = 4, f(2) + f''(2) = 13 and ∫ 0 1 f ( x ) d x = 5 3 \large \int_{0}^{1} f(x)\, dx = \dfrac{5}{3} .

Where p , q , r p, q, r are integers. Find p + q + r p+q+r .


The answer is 3.

Samara Simha Reddy by Samara Simha Reddy , 22, India

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1 solution

Since ∫ 0 1 f ( x ) d x = 5 3 \large \int_{0}^{1} f(x)\, dx = \frac{5}{3}

⟹ \implies p 3 + q 2 + r = 5 3 \large \frac{p}{3} + \frac{q}{2} + r = \frac{5}{3} .......... 1 \boxed1

f ′ ( 1 ) = 4 ⟹ 2 p + q = 4 \large f'(1) = 4 \implies 2p + q = 4 .......... 2 \boxed2

f ( 2 ) + f ′ ′ ( 2 ) = 13 \large f(2) + f''(2) = 13

⟹ ( 4 p + 2 q + r ) + 2 p = 13 \implies (4p + 2q + r) + 2p = 13

⟹ 6 p + 2 q + r = 13 \implies 6p + 2q + r = 13 .......... 3 \boxed3

By solving the equations 1 , 2 , 3 1, 2, 3 , we get

p = 2 , q = 0 , r = 1 \large p = 2, q = 0, r = 1

Therefore, p + q + r = 3 \large p + q + r = 3

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