`f(x) = (x + 1)/(sqrt(x^2 + 1))` (a) Use a graph of `f` to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of `x` at which `f` increases most...

`f(x) = (x + 1)/(sqrt(x^2 + 1))` (a) Use a graph of `f` to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of `x` at which `f` increases most rapidly. Then find the exact value.

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Textbook Question

Chapter 4, 4.3 - Problem 55 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`f(x)=(x+1)/sqrt(x^2+1)`

Now let us differentiate using quotient rule,

`f'(x)=(sqrt(x^2+1)d/dx(x+1)-(x+1)d/dxsqrt(x^2+1))/(x^2+1)`

`f'(x)=(sqrt(x^2+1)-(x+1)(1/2)(x^2+1)^(-1/2)(2x))/(x^2+1)`

`f'(x)=(sqrt(x^2+1)-((x(x+1))/sqrt(x^2+1)))/(x^2+1)`

`f'(x)=((x^2+1)-(x^2+x))/(x^2+1)^(3/2)`

`f'(x)=(1-x)/(x^2+1)^(3/2)`

Now to find the critical numbers, solve for x for f'(x)=0

`(1-x)/(x^2+1)^(3/2)=0`

x=1

f(x) increases most rapidly at x=1

`f(1)=(1+1)/sqrt(1^2+1)`

`f(1)=2/sqrt(2)=sqrt(2)`

Graph is attached and the function is maximum at x=1, f(x) is approx. 1.4

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