Question

If f(x) = 9.4x-24.2x and g(x) = cos2x- 4 sin^2X-Men 5cos^2 x, then the values of x and y for which f(x) + 12 = g (y)are​

Answer 1

If $f(x)=9.4^x-24.2^x$ and g(x) = cos2x - 4sin²x - 5cos²x

we have to find the values of x and y for which f(x) + 12 = g(y)

solution : $f(x)=9.4^x-24.2^x$

= $(3.2^x)^2-3(3.2^x)(4)+(4)^2-(4)^2$

= $(3.2^x-4)^2-4^2$

= $(3.2^x-4)^2-16$

then it is clear that, $(3.2^x-4)^2$ ≥ 0 for all x belongs to real numbers.

so f(x) ≥ - 16

⇒f(x) + 12 ≥ -12 .....(1)

again, g(x) = cos2x - 4sin²x - 5 cos²x

= 2cos²x - 1 - 4(1 - cos²x) - 5cos²x

= 2cos²x - 1 - 4 + 4 cos²x - 5cos²x

= cos²x - 5

or, g(y) = cos²y - 5

we know, cos²y ≥ 0 for all x belongs to R

so, g(y) ≤ - 4 for all x belongs to R. ......(2)

from equations (1) and (2) we get,

f(x) + 12 = g(y) only when f(x) + 12 = -4 = g(y)

so, cos²y - 5 = -4

⇒cos²y = 1 ⇒y = nπ , n ∈ I

and f(x) + 12 = -4

⇒(3.2^x - 4)² - 16 + 12 = -4

⇒(3.2^x - 4) = 0

⇒x = log₂4/3

Therefore the value of x = log₂(4/3) and y = nπ, where n ∈ I