Question

20. Find the maxima and minima of f(x,y) =
${x}^{3} {y}^{2} (12 - x - y)$
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Answer 1

Answer:

Consider the function f(x)={x}^{2}+1 over the interval (\text{−}\infty ,\infty ). As x\to \text{±}\infty , f(x)\to \infty . Therefore, the function does not have a largest value. However, since {x}^{2}+1\ge 1 for all real numbers x and {x}^{2}+1=1 when x=0, the function has a smallest value, 1, when x=0. We say that 1 is the absolute minimum of f(x)={x}^{2}+1 and it occurs at x=0. We say that f(x)={x}^{2}+1 does not have an absolute maximum (see the following figure).

Step-by-step explanation:

Let f be a function defined over an interval I and let c\in I. We say f has an absolute maximum on I at c if f(c)\ge f(x) for all x\in I. We say f has an absolute minimum on I at c if f(c)\le f(x) for all x\in I. If f has an absolute maximum on I at c or an absolute minimum on I at c, we say f has an absolute extremum on I at c.