Find any relative maxima and minima. Use a graphing utility to check your results

Question

Find any relative maxima and minima. Use a graphing utility to check your results

Y= e^x/2x

relative minimum (x,y) = ?

Answer 1

y = e^x/2x

Take the derivative set to 0 and find the relative minimum or maximum

y' = (2xe^x - 2e^x)/4x^{2 = 0}

y' = (xe^x - e^x)/4x² = 0

at x = 1 is the relative minimum

Answer 2

We know that any minima or maxima will occur when the derivate of the function is either zero or undefined. If we then calculate the derivative of the function utilizing the product rule, with f(x)=e^x and g(x)=1/(2x), we see that d/dx(f(x)g(x))=f'(x)g(x)+g'(x)f(x), so since f'(x)=e^x and g'(x)=-2/(2x)^2 utilizing the chain rule, we can plug in to find the derivative is e^x(1/(2x)-2/(2x)^2), so since e^x is always positive for any real number, to find minima or maxima we set 1/(2x)-2/(2x)^2 to be zero since the original function is undefined at zero. Thus, solving 1/(2x)-2/(2x)^2=0, we see that 2x-2=0 or x=1. Utilizing the first derivative test, since for x slightly larger than 1, this expression is positive and for x slightly smaller than 1, it is negative, so we know that this point must correspond to a local minimum. To find this point, we plug x=1 back into the original equation to get e/2.