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Maths-

General

Easy

Question

$text If end text y equals e to the power of 4 x end exponent plus 2 e to the power of negative x end exponent$ satisfies the relation $fraction numerator d cubed y over denominator d x cubed end fraction plus A fraction numerator d y over denominator d x end fraction plus B y equals 0$ then value of A and B respectively are:

–13, 14
–13, –12
–13, 12
12, –13

hint Hint:

In this question, we have to find the value of A and B. Firstly, we will find all the differential values and then substitute in the equation and equate it to find the required vaue.

The correct answer is: –13, –12

$y equals e to the power of 4 x end exponent plus 2 e to the power of negative x end exponent d i f f e r e n t i a t i n g comma fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction e to the power of 4 x end exponent plus fraction numerator d over denominator d x end fraction 2 e to the power of negative x end exponent space space space space space space space equals 4 e to the power of 4 x end exponent minus 2 e to the power of negative x end exponent A g a i n space d i f f e r e n t i a t i n g comma fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator d over denominator d x end fraction 4 e to the power of 4 x end exponent minus fraction numerator d over denominator d x end fraction 2 e to the power of negative x end exponent space space space space space space space space space equals 16 e to the power of 4 x end exponent plus 2 e to the power of negative x end exponent A g a i n space d i f f e r e n t i a t i n g comma fraction numerator d cubed y over denominator d y cubed end fraction equals fraction numerator d over denominator d x end fraction 16 e to the power of 4 x end exponent plus fraction numerator d over denominator d x end fraction 2 e to the power of negative x end exponent space space space space space space space space space equals 64 e to the power of 4 x end exponent minus 2 e to the power of negative x end exponent s u b s t i t u t i b g space t h e s e space d e r i v a t i o n s space i n comma fraction numerator d cubed y over denominator d y cubed end fraction plus A fraction numerator d squared y over denominator d x squared end fraction plus B y equals 0 rightwards double arrow 64 e to the power of 4 x end exponent minus 2 e to the power of negative x end exponent plus A open parentheses 16 e to the power of 4 x end exponent plus 2 e to the power of negative x end exponent close parentheses plus B open parentheses e to the power of 4 x end exponent plus 2 e to the power of negative x end exponent close parentheses equals 0 rightwards double arrow e to the power of 4 x end exponent open parentheses 64 plus 4 A plus B close parentheses plus e to the power of negative x end exponent left parenthesis negative 2 minus 2 A plus 2 B right parenthesis equals 0 O n l y space p o s s i b l e space w h e n space t h e space t e r m space i s space p a r e n t h e s i s space i s space 0. rightwards double arrow 4 A plus B equals negative 64 space a n d space 2 B minus 2 A equals 2 rightwards double arrow A equals negative 13 space a n d space B equals negative 12$

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