Respuesta :
[tex]\bf a-b=18\implies a=18+b
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\textit{now, their product is }f=a\cdot b\implies f(b)=(18+b)b
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f(b)=18b+b^2
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\cfrac{df}{db}=18+2b\implies 0=18+2b\implies 0=9+b\implies \boxed{-9=b}[/tex]
so, that's our only critical point, run a first-derivative test on it, and you'll notice is a minimum
so, that's our only critical point, run a first-derivative test on it, and you'll notice is a minimum
Answer: The minimum product is -81.
Step-by-step explanation:
Let the larger number is x,
Then the smaller number will be (x-18) ( because the sum of these number is 18)
Hence, the product of these numbers,
f(x) = x(x-18)
⇒ [tex]f(x) = x^2-18[/tex]
By differentiating with respect to x,
f'(x) = 2x - 18
For maxima or minima,
f'(x) = 0
⇒ 2x - 18 = 0
⇒ 2x = 18
⇒ x = 9
Again differentiating f'(x) with respect to x,
We get, f''(x) = 2
At x = 9, f''(x) = Positive
Hence, for x = 9,
f(x) is minimum,
And, the minimum value of f(x) = f(9) = 9(9-18)=9(-9) = -81
Hence, the minimum product is -81.
