Counterexample: Suppose I'm selling a product with zero cost and I've determined the profit-maximizing price to be $1.00. This means profit as a function of price is maximal and therefore the slope is zero, i.e. raising the price by 1% makes sales go down by 1% and vice versa.
Now suppose my cost to produce goes from zero to $0.99 per unit, which drops my profit to $0.01. Now that 1% increase in price that only drops sales by 1% is well worth it because my profit per unit doubles. The profit-maximizing price is now likely well over $1.
Now suppose my cost to produce goes from zero to $0.99 per unit, which drops my profit to $0.01. Now that 1% increase in price that only drops sales by 1% is well worth it because my profit per unit doubles. The profit-maximizing price is now likely well over $1.