Maximum power in this resistor circuit

[Physics lover]

Homework Statement

The question is in Attempt at a solution

Relevant Equations

V=IR
P=V^2/R
P=I^2×R

My attempt-:

So the resistance is coming to be negative.Can anyone tell me my mistake.Thanks.

 

[trurle]

Homework Statement: The question is in Attempt at a solution
Homework Equations: V=IR
P=V^2/R
P=I^2×R

View attachment 249522

My attempt-:
View attachment 249523

So the resistance is coming to be negative.Can anyone tell me my mistake.Thanks.

Formally your answer is correct. Input source impedance is (3||6 Ohm)=2 Ohm, therefore from matching theorem your load must be also 2 Ohm for maximal transferred power. You just need to trim negative resistance answer to closest physical value (i.e. 0).

 

[haruspex]

So the resistance is coming to be negative.Can anyone tell me my mistake.Thanks.

Your error is in assuming all minima are local minima, i.e. where a gradient is zero.
Sometimes the global minimum occurs at one end of the valid range. In general, you should check the extremities of the range even if you do find a local minimum.

 

[Physics lover]

Your error is in assuming all minima are local minima, i.e. where a gradient is zero.
Sometimes the global minimum occurs at one end of the valid range. In general, you should check the extremities of the range even if you do find a local minimum.

Can you please explain me how can i do that here means how can i find the range of R.

 

[haruspex]

Can you please explain me how can i do that here means how can i find the range of R.

As @trurle posted, it cannot be negative.

 

[Delta2]

Can you please explain me how can i do that here means how can i find the range of R.

The range of R should either be given by the problem(for example in this problem it could have been given that the variable resistor is in the range for example $[4,404]$) or can be logically inferred like for example in this problem the range of R is logically inferred to be $[0,+\infty)$. So you look for maximum or minimum at the end points and how to prove that the value at the end point is a maximum (or minimum).

 

[DEvens]

As @trurle posted, it cannot be negative.

You worked out the derivative. You looked for zeros. Good. Now check the value of the function and the derivative at the end points.

Suppose the derivative is positive at R=0. Can the maximum be at R=0? Suppose the derivative is negative at R=0. How about 0 as a maximum then?

You should also think about a maximum R. Though I don't see one listed in your problem statement. Variable resistors have a maximum value. You should check what is happening for large R.

 

[Physics lover]

ok thanks to everyone.I understood it