While preparing a bacterial culture, if one can provide only one of either controlled temperature or continuos shaking, which one yields good growth of bacteria
Both temperature and continuous shaking are required for good growth of bacteria.
Temperature is more important. An increase in temperature will increase enzyme activity. But if temperatures get too high, enzyme activity will diminish, and the protein (the enzyme) will denature. On the other hand, lowering temperature will decrease enzyme activity. Every bacterial species has specific growth temperature requirements which is largely determined by the temperature requirements of its enzymes.
Even continuous shaking is required. Normally cultures shake at 150 - 250 rpm. This will help in aeration and will increase the density of the culture. Moreover, it will prevent bacteria from settling at the bottom due to gravity and the nutrients in culture will also be readily available to the bacteria for optimal growth.
Which factor is most important depends on the strain of bacteria. If oxygen is necessary for growth, then shaking is very important. If the bacteria tend to clump together, then shaking is important.
You'll just have to test and see how well your bacteria grows in the conditions you have available. You might have to get creative with equipment - like a rotostat to rotate or a heating pad to warm an area.
If you are growing E. coli, I'd use shaking in a temperature as close to body temperature as you can get. It will grow at room temperature too, just more slowly.
Both are important. Setting the right temperature is essential for enzyme activity, but the decision to keep shaking depends on your experimental, the type of bacteria, and so on.
That depends on what bacterial group you’re working with but generally temperature and shaking are important factors you can’t disregard whenever you are dealing with bacterial growth
Temperature will have the biggest effect on growth rate at low cell densities. Shaking speed (or aeration is what you are actually changing) will have a substantial effect on the final density your culture reaches. This can be impacted by either the rate of shaking, the type of vessel (baffled flask vs not baffled) and the volume of liquid in a flask. If you fill the flask with liquid to 25% or more of total volume, you need high speed shaking. If you fill the flask with 10% or less, then you can get good aeration at lower speed.
So to simplify the answer, temperature will control what is the maximum possible growth rate for that strain and media, aeration will affect whether or not you reach the maximum.