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Geometry Level 5

Let f ( x ) = 1 + 3 x 2 + 5 x 4 + 7 x 6 + ⋯ + 21 x 20 , x ∈ R f(x)= 1+3x^2 + 5x^4 + 7x^6+ \cdots + 21x^{20} , \quad x \in \mathbb{R}

and g ( x ) = − x 2 + 4 cos ⁡ 2 b − 4 sin ⁡ b − 7 , b ∈ R g(x) = -x^2 + 4 \cos^2b- 4 \sin b -7, \quad b \in \mathbb{R}

If d d is the shortest distance between f ( x ) f(x) and g ( x ) g(x) and d 1 , d 2 d_1,d_2 are the greatest and least values of d d respectively, then d 1 d 2 = ? \dfrac {d_1}{d_2}=?


The answer is 4.

Avadhoot Sinkar by Avadhoot Sinkar , 22, India

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1 solution

Avadhoot Sinkar
May 16, 2016

We see that the function f(x) is an even function with its value at x=0 as 1. Also the function goes on increasing on both sides of 0. While the function g(x) is an inverted parabola with its vertex on y axis and y coordinate being determined by b. So get the minimum and maximum value of trigonometric function in g(x) and thus we get d2=3 and d1=12 so d1/d2=4.

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