01 - 04 Number Problems in Maxima and Minima
Problem 1
What number exceeds its square by the maximum amount?
Solution 1
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Let
x = the number and
x2 = the square of the number
y = the difference between x and x2
x = the number and
x2 = the square of the number
y = the difference between x and x2
$y = x - x^2$
$y' = 1 - 2x = 0$
$x = \frac{1}{2}$ answer
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Application of Maxima and Minima
As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as $A = f(x)$. The common task here is to find the value of $x$ that will give a maximum value of $A$. To find this value, we set $dA/dx = 0$.
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Maxima and Minima | Applications
Graph of the Function y = f(x)
The graph of a function y = f(x) may be plotted using Differential Calculus. Consider the graph shown below.
As x increases, the curve rises if the slope is positive, as of arc AB; it falls if the slope is negative, as of arc BC.
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