Answer to Question #148338 in Calculus for Angelo

Question #148338
Find the number in the interval (1/3, 2) such that the sum of the number and its reciprocal
is a maximum.
1
Expert's answer
2020-12-10T14:17:59-0500
"Solution"

Let us consider "x"  be the number

And its reciprocal is "\\frac1x"

Now the sum of the number and its reciprocal is "x+\\frac1x"

Let "f(x)=x+\\frac1x"  on "[\\frac13, 2]"

Clearly f  is a rational function continues on "[\\frac13, 2]"

Now we need to find the absolute maximum (large) and absolute minimum (small) values of "f(x)" on "[\\frac13, 2]"


A continuous function "f" on a closed interval "[a,b]" has both absolute maximum and minimum values. These are obtained by using following procedure:

(i) First all critical numbers of the function are found on"[a,b]" 

(ii) Then "f"  is evaluated at the critical numbers and the end points "a" and "b"

The largest and the smallest values among the values of the function computed above gives the absolute maximum and absolute minimum values of the function.


Differentiating "f(x)" the expression for  with respect to "x"


"f'(x)=\\frac{d}{dx}\\begin{bmatrix}\nx+\\frac{1}{x}\n\\end{bmatrix}\\\\\n\\frac{d(x)}{dx}+\\frac{d(\\frac{1}{x})}{dx}\\ since\\begin{pmatrix}\n\\frac{d(x^n)}{dx}=nx^{n-1}\n\\end{pmatrix}\\\\\n=1-\\frac{1}{x^2}"


Since "f'(x)" is defined for all "x" in the given "[\\frac{1}{3},2]"

The critical numbers are obtained by equating the first derivative to 0.

Thus 


"f'(x)=0\\\\\n\\implies 1-\\frac{1}{x^2}=0\\\\\n\\implies \\frac{1}{x^2}=1"


Out of these two critical numbers "x=-1" does not belong to the given "[\\frac13,2]". Hence "x=-1" is discarded.

Therefore the lone critical number for the given function in the given

"[\\frac13,2]" is "x=1" .


Next the value of the function at the critical number and the end points is computed. These are

"f(\\frac{1}{3})=\\frac{1}{3}+3=\\frac{10}{3}\\\\\nf(1)=1+1=2\\\\\nf(2)=1+\\frac{1}{2}=\\frac{5}{5}\\\\"

Therefore  "f" attains the larger value at "x=\\frac13"  in the given"[\\frac13, 2]".


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