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Updated on: 12 Oct 2016, 03:58
Originally posted by MathRevolution on 12 Oct 2016, 03:31.
Last edited by Bunuel on 12 Oct 2016, 03:58, edited 1 time in total.
Last edited by Bunuel on 12 Oct 2016, 03:58, edited 1 time in total.
Renamed the topic and edited the question.
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E
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It’s well known that a company's profit=20,000x-25x^2, where x is the number of workers. If the profit of the company is the greatest, what is the value of x?
A. 50
B. 100
C. 150
D. 250
E. 400
A. 50
B. 100
C. 150
D. 250
E. 400
12 Nov 2016, 06:34
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MathRevolution
Notice the given formula P = 20,000x - 25x^2 is a quadratic function. Furthermore, the graph of this function will be a parabola opening downward, and the vertex of this parabola will be the maximum point of the graph. Since we need to determine a value of x in which the company makes the greatest profit, we need to determine the value of x at the vertex of the parabola.
Recall that x = -b/(2a) is the formula to find the x-value of the vertex; therefore, the number of workers that maximizes the profit is:
x = -20,000/[2(-25)] = -20,000/-50 = 400
Answer: E
12 Oct 2016, 16:40
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Alternate approach:
To find Max or Min of a polynomial f(x), Equate f'(x)=0
f(x) = 20,000x-25x^2
Differentiate f(x)
f'(x) = 20,000 - 25*2*x = 0
50x = 20,000
x=400
D
Note: To confirm whether x is maximum or minimum, substitute x in f''(x)
If f''(x) <0, then f(x) is maximum
If f''(x) >0, then f(x) is minimum
In this case, f''(x)= -50 < 0
To find Max or Min of a polynomial f(x), Equate f'(x)=0
f(x) = 20,000x-25x^2
Differentiate f(x)
f'(x) = 20,000 - 25*2*x = 0
50x = 20,000
x=400
D
Note: To confirm whether x is maximum or minimum, substitute x in f''(x)
If f''(x) <0, then f(x) is maximum
If f''(x) >0, then f(x) is minimum
In this case, f''(x)= -50 < 0
