This is an incorrect statement in any market structure. The way to maximize profit (or minimize loss) is to sell your output at the price that is related to the point where your marginal revenue is equal to your marginal costs.
If you think about a firm that holds a monopoly, you will see why this is true. Let's say your firm has a monopoly on cable TV in some area (and let's assume satellite TV does not exist). Can you put the price up as high as you want? Of course not. People would simply not buy cable TV (if the price were high enough).
In all market structures, you set the price at a point on the demand curve that corresponds to the quantity of production where marginal revenue equals marginal costs.
Most certainly selling outputs of a company at the highest price will not result in highest profit. As a matter of fact trying to sell product at very high prices may result in the highest losses rather than highest profits.
The profit of a company is the surplus of the total revenue over the total cost. A high price will give a company a high revenue per unit sold. But a high price is also most likely to reduce the demand and hence the total number of unit sold. In this way maximum total revenue may increase with increase in price, but after certain point level of price it begins to fall. The firm is likely to make maximum profit at a point where the price is equal to or less than the price that corresponds to the price that gives maximum revenue.
Even the price that gives the maximum revenue will not give maximum profit if this price is less than the marginal cost of the product. In such cases the company's saving in cost by reducing total revenue is more than the amount of total revenue lost. This means that surplus of total revenue over total cost, which represents the profit, will increase by reducing total revenue.