In a certain company, the formula for maximizing profits is

Couple of things:

Quadratic expression \(ax^2+bx+c\) reaches its extreme values when \(x=-\frac{b}{2a}\). When \(a>0\) extreme value is minimum value of \(ax^2+bx+c\) (maximum value is not limited) and when \(a<0\) extreme value is maximum value of \(ax^2+bx+c\) (minimum value is not limited).

You can look at this geometrically: \(y=ax^2+bx+c\) when graphed on XY plane gives parabola. When \(a>0\), the parabola opens upward and minimum value of \(ax^2+bx+c\) is y-coordinate of vertex, when \(a<0\), the parabola opens downward and maximum value of \(ax^2+bx+c\) is y-coordinate of vertex.

Examples:
Expression \(5x^2-10x+20\) reaches its minimum when \(x=-\frac{b}{2a}=-\frac{-10}{2*5}=1\), so minimum value is \(5x^2-10x+20=5*1^2-10*1+20=15\).

Expression \(-5x^2-10x+20\) reaches its maximum when \(x=-\frac{b}{2a}=-\frac{-10}{2*(-5)}=-1\), so maximum value is \(-5x^2-10x+20=-5*(-1)^2-10*(-1)+20=25\).

Back to the original question:

In a certain company, the formula for maximizing profits is P = -25x^2 + 7500x, where P is profit and x is the number of machines the company operates in its factory. What value for x will maximize P?

A) 10
B) 50
C) 150
D) 200
E) 300

\(P=-25x^2+7500x\) reaches its maximum (as \(a=-25<0\)) when \(x=-\frac{b}{2a}=-\frac{7500}{2*(-25)}=150\).

Answer: C.

As for the minimum value of P: minimum value of P is not limited (x increase to +infinity --> P decreases to -infinity).

Hope it helps.
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If you understand Calculus, this is the easy way of solving it. Nobody else mentioned, but you also need to take the second derivative.

\(f(x) = -25x^2 + 7500x\)

\(f'(x) = -50x + 7500\)

\(-50x + 7500 = 0\)
\(7500 = 50x\)
\(x = 150\)

\(f''(x) = -50\)

If the second derivative is negative, it is a maximum. If it is positive, it is a minimum.

Pertaining to your question,
What value for x will maximize P? if you get the derivative of P = -25×2 + 7500x then the equation will bcome,

-50X + 7500 = 0 ( equating to 0 to get max value)
X = 150

People, please write the equation properly.

I was wondering how p = -50 + 7500x we can solve this question for 5 mins
It is simply as increasing functions.

Please edit the question and write P = \(-25 * x^2\) + \(7500x\)

We can solve the above problem by calculas Maxima and minima or by using the perfect score approach.

If you're not comfortable with calculus, here is how I would do it.

Recognize that 25 is a factor of 7500. If we take this out, we have two parts to the equation:

-X^2 & 300X

One part of the equation brings our value down, whereas the other part brings our value up. At this point, we can test the numbers in the answer choice. Notice that they are very straight forward to square, and multiplication by 300 is very easy.



A) 10 - 3000 - 100 = 2900
B) 50 - 15000 - 2500 = 12500
C) 150 - 45000 - 22500 = 22500
D) 200 - 60000 - 40000 = 20000
E) 300 - recognize that this is zero

Therefore, answer is 150.

If you are able to spot the 25 in 7500, you can go through this process in and around 2 minutes. Also, if you tested C or D first, you'd recognize that A and B would never be able to reach their output, and spotting that E makes the equation equal to 0, there is only 2 numbers you would need to test.

Hope this helps those - like myself - who haven't thought about calculus for over half a decade.

Why does P needs to be set to 0 in order to get max value for P. I'm totally lost here.
P = -25×2 + 7500x
If x=150, then P=1124950
If x=300, then P=2249950
2249950>1124950, so P is greater when x=300 vs. x=150.

Why is x=150 the correct answer?

Not p=0 its dP/dx =firt derviative of P with respect to x
dP/dx =0 = -50x +7500 =0
x=150
Please not that we treated x2= x^2 (please use exponential symbol for that)

P = -25×^2 + 7500x

If x=150, then P=562500
If x=300, then P=0


Do you have knowledge of calculus (Derviatives)..?.

I took calculus a LONG time ago, so I prob. forgot all of it already. I thought GMAT doesn't test calculus concepts? Why is this "GMAT" type problem requires calculus to solve?

Thanks for this post to have a recall of calculus..."What value for x will maximize P"
Need more clarification on minimum value....

what if require the value of x when profit is lowest???

My understanding is that it comes from the formula for minimize profit. This will be given in question stem... such as this is present here... "the formula for maximizing profits is P = -25x2 + 7500x"....

Please put the correct understanding here.

Hey Bro,
Thanks a lot..."If the second derivative is negative, it is a maximum. If it is positive, it is a minimum. "...

I was looking for this only....its really long long time I left calculus so was trying some lead...Kudos Given..

Hello

I had two points to share
1) Please confirm that I understand the equation in the question correctly : P=(-25)(2)+ (7500x). If I'm reading it correctly, then how does this equation turn into -50x+7500x?

2) How testable is this content on actual GMAT test? because I looked into the official guide testable topics as well as Manhattan books and it seems that calculus is not one of the topics. Where can I find this content to study?

______
Done. Thank you.
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My approach is just plugging ,
X = 150 , where p = 562500
X = 300 , p = 0 , x = 200 = 500000
so C

In a certain company, the formula for maximizing profits is P = -25x^2 + 7500x,, where P is profit and x is the number of machines the company operates in its factory. What value for x will maximize P?

A) 10
B) 50
C) 150
D) 200
E) 300

SOLUTION:

P = -25x^2 + 7500x
P = 25x (-x + 300) ----- (I)

Plug in answer choices in (I):

A) 25*10 (290)
B) 25*10*5 (250)
C) 25*10*15 (150)
D) 25*10*20 (100)
E) 25*10*30 (0)

Dividing A-E by 250:
A) 290
B)1250
C)2250
D)2000
E) 0

ANSWER: C

Firstly I am not sure derivative will be useful in GMAT or not ..

here is the solution though

D(p)/D(x) = -50x+7500
Now the general rule is to equate the derivative to zero to get the values of any maxima or minima
=> 50x=7500
=> x=150

Kudos if you like my solution ..It helps..
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Differentiation is not required for GMAT, but if you know it , then there is no harm in applying it. But for the sake of people who are not aware of differentiation, for all these max/min question involving quadratic equations, the best strategy is to come up with perfect squares of the form \((a \pm b)^2\) and then maximize or minimize by remembering the \((a \pm b)^2 \geq 0\) as shown below:

P = \(-25x^2 + 7500x\) ---> P = \(-25 (x^2-300x)\) = \(-25 (x^2-2*150*x+150^2 - 150^2)\)= \(-25 (x^2-2*150*x+150^2)+25*150^2\)= \(-25 (x-150)^2 + 25*150^2\) = a negative quantity + positive quantity.

Thus to maximize P, you need to minimize the perfect square \((x-150)^2\) and the minimum value of a perfect square = 0 ---> For P to be maximized , \((x-150)^2 =0\) ---> \(x=150\).

Any other value of x will only reduce P

Hence C is the correct answer.

Hope this helps.
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First we can see that the graph of P = -25x^2 + 7500x is a down-opening parabola. This means that the vertex of the parabola will be at the maximum value. We find the x-coordinate of the vertex using the following equation:

x = -b/(2a)

x = -7500/(2 x -25) = -7500/-50 = 150

The maximum profit will be realized when 150 machines are sold.

Answer: C
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Here's how I did it...

\(P = -25x^2 + 7500x = -25x(x - 300)\)

For \(P\) to be positive, \(x\) must be less than 300, so option E is out.

(D): \(-25(200) (-100) = 500,000\)
(C): \(-25(150) (-150) = 562,500\)
(B): \(-25(50) (-250) = 312,500\)
(A): \(-25(10) (-290) = 72,500\)

\(P\) is maximum when \(x = 150\), so option C.

Using calculus:
\(-25x^2 + 7500x\) is maximum when \(\frac{d}{dx}(-25x^2 + 7500x) = 0\).

\(\frac{d}{dx}(-25x^2 + 7500x) = 0\)

\(-50x + 7500 = 0\)

\(50x = 7500\)

\(x = \frac{7500}{50} = 150.\)
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What I did was first solve for what values minimise the profit. By solving for =0, we find that x is either 0 or 300.

BUT, these values minimise the profit, however we want to maximise. So with a quick look at the possible answers, the "middle" value between these two extremes that minimise the profit, will maximise it.

Not very mathematically correct, I guess, but works. No?