Question 511607
x + y = 26
so
x = 26-y
.
The product is 
z = x * y = (26-y) * y
.
z = 26y -y^2
.
To find the maximum, take the first derivative and solve it.
.
dz/dy = 26 - 2y
.
26-2y = 0
26 = 2y
y = 13
.
Check:
<table>
<tr><td>y&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<td>x=26-y&nbsp;&nbsp;&nbsp;<td>x*y&nbsp;&nbsp;&nbsp;</tr>
<tr><td>13<td>13<td>169</tr>
<tr><td>12<td>14<td>168</tr>
<tr><td>11<td>15<td>165</tr>
<tr><td>10<td>16<td>160</tr>
<tr><td>9<td>17<td>153</tr>
<tr><td>8<td>18<td>144</tr>
<tr><td>7<td>19<td>133</tr>
<tr><td>6<td>20<td>120</tr>
<tr><td>5<td>21<td>105</tr>
<tr><td>4<td>22<td>88</tr>
<tr><td>3<td>23<td>69</tr>
<tr><td>2<td>24<td>48</tr>
<tr><td>1<td>25<td>25</tr>
</table>
.
Answer:  The pair of numbers are 13 and 13 that total 26 and maximizes the product.
.
Done.