Some questions I need help with =] Watch

Mayonaise
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#1
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Hey can anyone help me out?

How would I go about figuring out:

What is the minimum value of 4cos( x - \frac{\pi}{3}) + 6?



And also:

A function f is given by  f(x)=(x-2)^2 -3

The function g is given by  g(x)= \frac{1}{f(x) + 10}

Which of the following statements about the stationary value of g is true?

A. minimum value of g is 7
B. maximum value of g is 7
C. min value of g is  \frac{1}{7}
D. max value of g is  \frac{1}{7}


Thankyou
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generalebriety
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What's the minimum value of cos(anything)? So what's the minimum value of 4cos(...) + 6?

Does f have a minimum point or a maximum point? (Think about the minimum value of a^2.) What's its value? Will this value of x give a minimum or a maximum point of g? What's its value?
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turgon
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Anyway for the first bit, differentiate the function, equal that to zero and solve x. Then find the second derivative and substitute your values for x in; you need the value for x which gives a positive number. Substitute that value in the orginal equation to get the minimum value of that function.

(Better still listen to generalebriety, I m talking rubbish)
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generalebriety
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(Original post by turgon)
Anyway for the first bit, differentiate the function using the product rule, equal that to zero and solve x. Then find the second derivative and substitute your values for x in; you need the value for x which gives a positive number. Substitute that value in the orginal equation to get the minimum value of that function.
Why bother? It's in completed square form. It's really easy to find the stationary point.
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Mayonaise
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I want to say the minimum value for cos(..) is -1?? =S
If the min value for cos is -1, would that make 2?

I'm sorry, I really don't have a clue about the second one??
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generalebriety
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(Original post by Mayonaise)
I want to say the minimum value for cos(..) is -1?? =S
If the min value for cos is -1, would that make 2?
That's right.

(Original post by Mayonaise)
I'm sorry, I really don't have a clue about the second one??
Does (anything)^2 have a minimum value or a maximum value? And what is it? (Think of the graph of y = x^2.) So if f(x) = (something)^2 - 3, what's its (minimum/maximum) value? Does this mean that g has a minimum or a maximum value, and what is it?
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turgon
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Yes the first bit does equal to 2. In the second bit think of a value for x which would make (x-2)^2 = 0 and hence the minimum value for f(x).
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Mayonaise
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Um is y=x^2 a parabula that passes through the origin?
Does that mean y=(x-2)^2 - 3 would just move two places to the right and down three?
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Mayonaise
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So would the min value be -3, and put that into
Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
g(x)=\frac{1}{f(x)+10)
which would give me \frac{1}{7}

Is that right?
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generalebriety
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(Original post by Mayonaise)
So would the min value be -3, and put that into
Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
g(x)=\frac{1}{f(x)+10)
which would give me \frac{1}{7}

Is that right?
Yep. Now, is that a maximum or a minimum (of g)?
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Mayonaise
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Um I'm not sure, how do I tell?
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generalebriety
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(Original post by Mayonaise)
Um I'm not sure, how do I tell?
Well, is it a maximum or minimum of f?

Reciprocation turns maxima into minima and vice-versa. 10 is bigger than 5, but 1/10 is less than 1/5.
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Mayonaise
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Is it a minimum of f? Is g(x) just f(x) flipped?

EDIT: No wait, a maximum? lol
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generalebriety
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(Original post by Mayonaise)
Is it a minimum of f? Is g(x) just f(x) flipped?

EDIT: No wait, a maximum? lol
Your value is -3.

Does (x-2)^2 - 3 have a minimum point or a maximum point? Think of the graph.
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Mayonaise
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It has a minimum tp. Does that make g have a max tp?
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generalebriety
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(Original post by Mayonaise)
It has a minimum tp. Does that make g have a max tp?
Yep. Do you understand why?
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Mayonaise
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Does the parabula flip at the f(x)'s tp making g(x)'s tp a maximum?
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Mayonaise
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Oh wait no, why would g's tp be 1/7?
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generalebriety
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(Original post by Mayonaise)
Does the parabula flip at the f(x)'s tp making g(x)'s tp a maximum?
Kind of, yes. Basically, if the variable u gets closer to 0, the variable (1/u) gets further from 0. Look at what happens to the graph of y = 1/x as x gets close to zero - it goes off to infinity at either end. So a minimum of f(x) will be a minimum of f(x) + 10 (obvious translation of the graph up ten units), which will be a maximum of g(x) (because big numbers get translated into small ones by reciprocation, and vice-versa).
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generalebriety
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(Original post by Mayonaise)
Oh wait no, why would g's tp be 1/7?
If it worries you, differentiate g.
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