Final answer:
To find the absolute minimum value of the function g(x) = e^x / 4x, we can take the derivative, set it equal to zero to find the critical points, and examine the behavior of the function. The absolute minimum value is 0, which occurs at x = 1.
Step-by-step explanation:
To find the absolute minimum value of the function g(x) = e^x / 4x, we can first take the derivative of the function and set it equal to zero to find the critical points.
Taking the derivative, we get:
g'(x) = (e^x * 4x - e^x * 4) / (4x)^2
Setting g'(x) equal to zero, we have:
e^x * 4x - e^x * 4 = 0
Factoring out e^x * 4, we get:
(e^x * (4x - 4)) / (4x)^2 = 0
Simplifying, we have:
e^x * (4x - 4) = 0
Since e^x is always positive and never equal to zero, the only critical point is when 4x - 4 = 0. Solving for x, we get x = 1.
To determine if this critical point is a minimum or maximum, we can examine the behavior of the function as x approaches positive infinity and as x approaches positive zero.
As x approaches positive infinity, the function approaches zero since both the numerator and the denominator grow exponentially. As x approaches positive zero, the function also approaches zero since the numerator approaches 1 and the denominator approaches 1 as well.
Therefore, the absolute minimum value of the function is 0, which occurs at x = 1.