Respuesta :
Let's assume that number as x
so,
the sum of the number and its reciprocal is
[tex]S(x)=x+\frac{1}{x}[/tex]
Firstly, we will find derivative
[tex]S'(x)=1-\frac{1}{x^2}[/tex]
now, we can set it to 0
and then we can solve for x
[tex]S'(x)=1-\frac{1}{x^2}=0[/tex]
[tex]x=-1,x=1[/tex]
Since, only x=1 lies on [1/2,3/2]
so, we will consider only x=1
now, we can plug end values and x=1 into S
At x=1/2:
we can plug x=1/2
[tex]S(\frac{1}{2})=\frac{1}{2} +\frac{1}{\frac{1}{2}}[/tex]
[tex]S(\frac{1}{2})=\frac{1}{2} +2[/tex]
[tex]S(\frac{1}{2})=\frac{5}{2}[/tex]
At x=1:
we can plug x=1
[tex]S(1)=1 +\frac{1}{1}[/tex]
[tex]S(1)=1 +1[/tex]
[tex]S(1)=2[/tex]
At x=3/2:
we can plug x=3/2
[tex]S(\frac{3}{2})=\frac{3}{2} +\frac{1}{\frac{3}{2}}[/tex]
[tex]S(\frac{3}{2})=\frac{3}{2} +\frac{2}{3}[/tex]
[tex]S(\frac{3}{2})=\frac{13}{6}[/tex]
(a)
Smallest value:
The smallest value among them
[tex]S(1)=2[/tex]
So, a number is 1..........Answer
(b)
Largest value:
The largest value among them
[tex]S(\frac{1}{2})=\frac{5}{2}[/tex]
So, a number is [tex]\frac{1}{2}[/tex]........Answer
The sum of the number and its reciprocal is smallest when the number is 1 and the largest is when the number is 1/2.
What are maxima and minima?
Maxima and minima of a function are the extrema within the range, in other words, the maximum value of a function at a certain point is called maxima and the minimum value of a function at a certain point is called minima.
We have closed interval [1/2, 3,2]
Let's suppose the number is x
the sum of the number and it's reciprocal = [tex]\rm x +\frac{1}{x}[/tex]
Let's denote it with f(x)
[tex]\rm f(x) = x +\frac{1}{x}[/tex]
For the maxima and minima of the function first, we calculate its first derivative and equate it to zero.
[tex]\rm f'(x) = 1 -\frac{1}{x^2}[/tex]
f'(x) = 0
[tex]\rm 1 -\frac{1}{x^2}=0\\\\\rm x^2-1 = 0[/tex] (x ≠ 0)
x = ±1
Since the negative value does not lie on the given interval, we will take [tex]\rm x = 1[/tex]
Calculate the f(x) value at x = 1, x = 1/2, and x = 3/2
[tex]\rm f(1) = 1 +\frac{1}{1} \Rightarrow2[/tex]
[tex]\rm f(\frac{1}{2} ) = \frac{1}{2} +\frac{1}{\frac{1}{2} }\Rightarrow \frac{5}{2}[/tex]
[tex]\rm f(\frac{3}{2} ) = \frac{3}{2} +\frac{1}{\frac{3}{2} }\Rightarrow \frac{13}{6}[/tex]
Thus, the sum of the number and its reciprocal is smallest when the number is 1 and the largest is when the number is 1/2.
Know more about the maxima and minima here:
brainly.com/question/6422517
