What is the point of output that will minimise the average total cost? answers should be supported by a digram

I an not sure if it is possible to draw diagrams on this forum of question and answer. However, I will answer your question in words and some simple mathematical expressions.

First of all let us understand the nature of costs involved in producing any item. Let us say we are producing toothbrushes on a simple tooth brush making machine.

Irrespective of number of toothbrush produced in a day we will incur some fixed costs such as rent for the place where machine is installed, depreciation on machine, interest on the money invested in the manufacturing facility, wages of the person operating the machine. This is the fixed cost of production which remains constant irrespective of the number of toothbrushes produced from the machine. As we increase the production of toothbrushes it gets distributed over increasing number of toothbrushes. If we draw a graph of fixed cost on Y-axis against the total production on X-axis it will be a downward sloping curve tending to meet X-axis  for very large production level.

In addition to the fixed costs, we also have a variable component of the cost. This cost varies along with the quantity of toothbrushes produced. More toothbrushes we produce higher is the variable cost. This is the cost of inputs like raw material, electricity, and  machine repair. This is called variable cost.

We can have two types of behavior for the variable cost. The variable cost per additional tooth brush produced remains same irrespective of the total number of tooth brushes produced. That is the marginal cost of producing every toothbrush remains same across the whole range of production level. If we draw a graph of total quantity produced on X-axis and marginal variable cost on Y-axis, it will be a horizontal straight line. In this case the marginal variable cost in all cases is same as the average variable cost

In the other alternative the marginal variable cost depends on the total level of production. Typically the marginal variable cost as production rises, till it reaches a minimum level, after which the cost begins to rise. Thus the marginal cost graph for this type of variable cost will be a U-shaped curve.

Having understood the basic cost behavior we will now determine level of production at which the average cost will be lowest.

We will first take the case where marginal variable cosy is constant. In this case the total cost and average cost of production are given by the following equations.:

Total Cost  = F + N*v

Average Cost = Fixed Cost/N = F/N + v

where F = fixed Cost; N = Number of toothbrushes produced; and v = variable cost per unit (this is fixed).

From this equation it is clear that the average cost will go on reducing as the total quantity produced is increased. Thus, theoretically the point of minimum average cost lies at infinity.

Taking the case of when marginal cost of production varies, it is clear that the average cost will definitely reduce till the minimum marginal cost point is reached. After that point the Fixed cost component of average cost (i.e. F/N) will reduce the average cost. At the same time increasing marginal cost will increase the average cost. Initially the effect of decreasing  average fixed cost will be higher than that of increasing average marginal cost, but as production level increases the two will become equal. This is the level at which the total average cost will be the lowest.