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"Marginal cost is never equal to variable costs." Please explain why this is false.

Marginal cost will be equal to variable cost when the number of units produced is 1. Thus, the statement "marginal cost is never equal to variable cost" is false.

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To understand why this statement is true for every situation after the first unit is produced, it is necessary to understand exactly what variable cost is. As the name implies, variable cost varies with the level of output produced. For instance, in a bakery, to produce one incremental unit of bread requires some additional flour, eggs and water. Thus, the total costs of these items will vary with each additional production of units, or loaves of bread.

Let’s say the unitary variable cost of these ingredients for the production of one loaf is $1.00. If the entire bag of flour costs $10 and each bag has enough flour to produce 20 loaves, then the flour component of each unitary cost of production is $0.50 ($10 divided by 20 loaves). The $1.00 variable cost covers the cost of all of these ingredients together plus any other ingredient (salt, for example) required to produce one unit.

Variable cost does not include costs such as rental of the bakery facilities, as that is a fixed amount...

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Marginal cost function is the first derivative of the cost function.

Marginal cost expresses how much are changing costs, when production of a good increases (usually with an infinitesimal unit). This value may be, of course, even negative. Marginal cost intersects the average costs always  their minimum point.If marginal costs decreasing, the point of intersection of two curves is the point of maximum average costs.

If marginal costs are higher than average costs, without fixed costs, is reached the minimum level of profitability. If a firm produces below the minimum threshold of profitability, is not good to produce more, because it not even cover variable costs.

It's better when marginal costs exceed average costs, including fixed costs. From this perspective, the optimal threshold of profitability, producer gets profit.

 Marginal costs formula(first derivative of cost function depending on x):