Inflexio!!

Calculus Level 3

Let a curve be defined as y = a x 4 + b x 3 + c x 2 + d x + e y = ax^4 + bx^3 + cx^2 + dx + e . If the condition for which it has points of inflection over the entire set of real numbers be

k b 2 − m a c > 0 kb^2 - mac> 0

where k and m are co-prime integers,

Find k + m k + m


The answer is 11.

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Vishwak Srinivasan by Vishwak Srinivasan , 24, India

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1 solution

The condition to obtain points of inflection is that

d 2 y d x 2 = 0 \displaystyle \dfrac{d^2y}{dx^2} = \boxed{0}

Double differentiating y = a x 4 + b x 3 + c x 2 + d x + e y = ax^4 + bx^3 + cx^2 + dx + e , we get

d 2 y d x 2 = 12 a x 2 + 6 b x + 2 c = 0 \dfrac{d^2y}{dx^2} = 12ax^2 + 6bx + 2c = 0

It is a quadratic equation that has real roots.

Hence

D > 0 D > 0

36 b 2 − 4 × 24 a c > 0 = 3 b 2 − 4 × 2 a c > 0 = 3 b 2 − 8 a c > 0 36b^2 - 4 \times 24ac > 0 = 3b^2 - 4\times 2ac > 0 = 3b^2 - 8ac > 0

k = 3 k = 3 and m = 8 m = 8

k + m = 11 k + m = \boxed{11}

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