Application of Derivatives

The combined resistance R of two resistors R 1 and R 2 (R 1 , R 2 > 0) is given by . If R 1 + R 2 = C (a constant), show that the maximum resistance R is obtained by choosing R 1 = R 2 . - Wired Faculty

Question

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The combined resistance R of two resistors R₁ and R₂ (R₁ , R₂ > 0) is given by $1 over straight R space equals space 1 over straight R subscript 1 plus 1 over straight R subscript 2$ .
If R₁ + R₂ = C (a constant), show that the maximum resistance R is obtained by choosing R₁ = R₂.

Solution

Here,    $1 over straight R space equals space 1 over straight R subscript 1 plus 1 over straight R subscript 2 space space space space space space space space space space rightwards double arrow space space space space 1 over straight R space equals space fraction numerator straight R subscript 1 plus straight R subscript 2 over denominator straight R subscript 1 space straight R subscript 2 end fraction$
   $rightwards double arrow space space space space 1 over straight R space equals space fraction numerator straight C over denominator straight R subscript 1 left parenthesis straight C minus straight R subscript 1 right parenthesis end fraction$ $open square brackets because space straight R subscript 1 plus straight R subscript 2 space equals space straight C close square brackets$
$rightwards double arrow space space space space space straight R space equals space fraction numerator straight R subscript 1 space left parenthesis straight C minus straight R subscript 1 right parenthesis over denominator straight C end fraction space space space space rightwards double arrow space space space straight R space equals space 1 over straight C left square bracket CR subscript 1 minus space straight R subscript 1 squared right square bracket space space rightwards double arrow space space dR over dR subscript 1 space equals space 1 over straight C left square bracket straight C minus 2 straight R subscript 1 right square bracket space space space dR over dR subscript 1 space equals 0 space space space space space space space space space space space space space rightwards double arrow space space space space 1 over straight C left square bracket straight C minus 2 straight R subscript 1 right square bracket space equals space 0 space space space space space space space rightwards double arrow space space space straight R subscript 1 space space equals straight C over 2 space space space space space space space space fraction numerator straight d squared straight R over denominator dR subscript 1 squared end fraction equals space 1 over straight C left square bracket negative 2 right square bracket space equals space minus 2 over straight C$
At    $straight R subscript 1 space equals space straight C over 2 fraction numerator straight d squared straight R over denominator dR subscript 1 squared end fraction space equals space minus 2 over straight C less than 0 space space space space space space rightwards double arrow space space space straight R space has space straight a space local space maximum space at space straight R subscript 1 space equals space straight C over 2$
But $straight R subscript 1 equals space straight C over 2 space is space the space only space extreme space point. space$
$therefore space space space straight R space is space maximum space when space straight R subscript 1 space equals space straight C over 2 comma space space straight R subscript 2 space equals space straight C minus straight C over 2 space equals straight C over 2 space straight i. straight e. space when space straight R subscript 1 space equals space straight R subscript 2$

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