Calculus Made Easy Exercise 10 Question 9

Question

(9) The efficiency u of an electric generator at different values of output x is expressed by the general equation:

$$u = \frac{x}{a + bx + cx^2}$$

where $a$ is a constant depending chiefly on the energy losses in the iron and $c$ a constant depending chiefly on the resistance of the copper parts. Find an expression for that value of the output at which the efficiency will be a maximum.

I am trying to solve this question but I am not sure how to go about approaching question. If you need I can show my steps so far. Please do not give away the answer but guide me in the right direction. That way I can learn more.

Many thanks and stay safe!

Answer 1

Hint:

Write $$u = \frac{1}{{a\over x} + b + cx}$$ so you have to find a minimum for ${a\over x} + cx$. This should be easy now, even without calculus...

Answer 2

Optimization problems generally follow the same procedure. Namely,

1. Determine your constraint ($\textit{i.e.} x < a, a \leq x \leq b, etc$)
2. Find the critical numbers
3. Determine whether each critical number is a max, min, or neither,
4. Evaluate your function at the relative extrema and endpoint (if any)
5. Choose the largest (or lowest) value