Determine the range of this trigonometric function

Calculus Level 2

If the range of cos ⁡ 4 x + sin ⁡ 2 x \cos^4 x + \sin^2 x for real x x is [ A , B ] [A,B] , then find the value of A + B A + B .


The answer is 1.75.

Vijay Simha by Vijay Simha , 47, USA

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1 solution

Marta Reece
Jun 27, 2017

Derivative − 4 cos ⁡ 3 x sin ⁡ x + 2 sin ⁡ x cos ⁡ x = 2 sin ⁡ x cos ⁡ x ( 1 − 2 cos ⁡ 2 x ) = 0 -4\cos^3x\sin x+2\sin x \cos x=2\sin x\cos x(1-2\cos^2x)=0

Solutions sin ⁡ x = 0 , cos ⁡ x = 0 \sin x=0, \cos x=0 produce locations where cos ⁡ 4 x + sin ⁡ 2 x = 1 \cos^4x+\sin^2x=1

Solution cos ⁡ 2 x = 1 2 \cos^2x=\frac12 produces cos ⁡ 4 x + sin ⁡ 2 x = 1 4 + ( 1 − 1 2 ) = 3 4 \cos^4x+\sin^2x=\frac14+(1-\frac12)=\frac34

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This is incomplete. You have only shown that 0 and 1 are the critical points, but you didn't show that they are absolute max/min points.

Pi Han Goh - 3 years, 11 months ago

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If all critical points of a periodic function are accounted for, and the values of the function in those points are 3 4 \frac 34 and 1 1 , then 3 4 \frac34 is the minimum and 1 1 is the maximum. If the function were not periodic, such as a third degree polynomial, for example, the function could escape those bounds, but for a periodic function this is not possible except at a critical point in which the derivative is not defined. There is not such point here.

Marta Reece - 3 years, 11 months ago

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