perplexus.info :: Just Math : Given Linear Sum, Find the Sum of Exponents

Given Linear Sum, Find the Sum of Exponents (Posted on 2024-05-19)

Each of x and y is a real number that satisfies this equation:
2x + y = -6

Determine the minimum value of this expression:

(1/4)^x + (1/2)^y

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Algebra and Calculus

| Comment 1 of 3

2x + y = -6

y = -6-2x Eqn 1

let z = (1/4)^x + (1/2)^y

(1/4)^x + (1/2)^(-6-2x) substitute from Eqn 1

(1/4)^x + 2^(6+2x)

(1/4)^x + 4^(3+x)

z = 4^(-x) + 4^(3+x)

dz/dx = -4^(-x)*ln(4) + 4^(3+x)*ln(4) = 0

4^(-x) = 4^(3+x)

-x = 3+x

x = -3/2

y = -6-2x = -3

(x,y) = (-3/2, -3)

Does 2x + y = -6 ? yes, check

(1/4)^x + (1/2)^y = 4^(3/2) + 2^3 = 16

Posted by Larry on 2024-05-19 08:23:16

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