Melody

The foci of an ellipse have to be inside the ellipse.  (0, 4) and (0, -4) are not in your ellipse.

\(\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right)\\ =\left(\frac{1}{b}+\frac{1}{a}\right)\left(\frac{1}{b}-\frac{1}{a}\right)\\ =\frac{1}{b^2}-\frac{1}{a^2}\\ smallest - biggest=smallest\\ \text{so I need the biggest }b^2 \text { and the smallest }a^2\\ biggest \;\;b^2=9\qquad smallest\;\;a^2=1\\ \text{smallest value = }\frac{1}{9}-1 = -\frac{8}{9}\\ \)

I would like to see someone answer this too.   

I shall try  

A perfect square is of the form x^2 where x is a positive whole number

One less than a perfect square is    x^2 -1

\(x^2 -1\)     is the difference of 2 squares.

difference means subtract and the two squares are   x^2    and    1^2

You need to memorize this factorization.

  \(\boxed{a^2-b^2=(a-b)(a+b)}\)

so        \(x^2-1^2=(x-1)(x+1)\)

If this is a prime number then the smaller factor must be 1

the factors are  (x-1) and (x+1)

since x is a positive whole number,  the smallest one will be x-1

therefore  x-1 must equal 1

so x must equal 2

Maxima and minima occur when the first derivative, (which is the gradient) is 0

So what is the first derivative of f(t)    ?      

Hint:  what is the opposite of integration?

Here is a graph that you might find useful.

Part a is easy.

Let

  \(z=e^{i\theta}=cos\theta + isin \theta\\ \bar{z}=cos\theta - isin \theta\\ \\~\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\times \frac{cos\theta - isin \theta}{cos\theta - isin \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{cos^2\theta + sin^2 \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{1}\\ \frac{1}{z}=cos\theta - isin \theta\\ \frac{1}{z}=\bar z\)

Same logic for w 

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999 divided by 37 = 27

plus it could be 000

makes a total of 28  just as guest already determined.     

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