Graph will make it easy

Calculus Level 4

e − n x = a sin ⁡ ( x ) + b cos ⁡ ( x ) \Large e^{-nx} = a\sin(x) + b\cos(x)

If there are infinite x x satisfying the above condition, then which of the following is true?

Note: n , a , b n, a, b are real numbers.

n > 0 , n < a , n < b n>0, n<a, n<b It only depends on a , b a,b n < 0 , n < a , n < b n<0, n<a, n<b It doesn't depend on n , a , b n,a,b

Akash Shukla by Akash Shukla , 26, India

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1 solution

Yashas Ravi
Nov 29, 2020

Since the midline of f ( x ) = a sin ⁡ ( x ) + b cos ⁡ ( x ) f(x) = a\sin(x) + b\cos(x) is 0 0 , and the function g ( x ) = e − n x g(x) = e^{-nx} approaches 0 0 , there will be an infinite value of solutions. Thus, there is no dependence on the values of n n , a a , and b b . However, if n = 0 n = 0 , then f ( x ) = 1 f(x) = 1 , so a 2 + b 2 ≥ 1 \sqrt{a^2 + b^2}≥1 since the amplitude of the trigonometric graph f ( x ) f(x) needs to be at least 1 1 for there to be infinite solutions.

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