Alan

The following 3d plot and contour plot might help:

Note also that the gradient in the x-direction is 2x - 4 and in the y-direction is 2y + 2, and that these gradients must be zero at the minimum.

Find the values of x and y from 2x - 4 = 0 and 2y + 2 = 0, then put these values back into your expression for temperature.

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a.  The x intercept is the value of x at which y = 0.   2x - 3*0 = -6, x=-3

b. The y intercept is the value of y at which x = 0.  2*0 - 3y = -6,  y = 2.

The equation can be rewritten as y = (2/3)x + 2

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Couldn't read your third one, but the first two are:

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4k² - 8 = 4k - 8

k² = k

k = 0 or k = 1

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Suppose he was going to paint each of rooms A, B and C a different colour.

1. If he decided that A= red, B=blue and C= green was a different combination from A=blue, B=green and C=red, then there would be smile478's 6840 possibilities.

2. However, the way the question is phrased, it sounds more like he would count these as the same combination of colours, in which case there would only be Chris's 1140 possibilities.

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Press the 2nd button, then choose asin

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If you really meant 2*7<18 then, as a Boolean expression, it is true.

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You got as far as a|6 - b| = 0.  This must mean that either a = 0 or b = 6.

If a = 0 then the function would reduce to f(x) = -1, a constant.  Since f(x) clearly isn't a constant we can't have a = 0, so we must have b = 6.

So the function is f(x) = a|x - 6| - 1

Use the point (0, 11) to find a.

11 = a*|0 - 6| - 1

11 = a*6 - 1

12 = a*6

a = 2

Finally,

f(x) = 2|x - 6| - 1

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1.  3x^(2/3)+4x^(1/3)-4=0

Let u = x^(1/3).  The equation above becomes

3u^2 + 4u - 4 = 0

This can be factored as

(3u - 2)(u + 2) = 0

so u = 2/3 and u = -2

Because u = x^(1/3) we must have x = u^3 so

x = (2/3)^3 = 8/27   and  x = (-2)^3 = -8

2. (1+1/x)^2 - 6(1+1/x) + 8 =0

Let u = 1+1/x. The equation becomes

u^2 - 6u + 8 = 0

This can be factored as

(u - 4)(u - 2) = 0

so u = 4 and u = 2

Since u = 1 + 1/x  we must have x = 1/(u-1) so

x = 1/(4 - 1) = 1/3  and  x = 1/(2 - 1) = 1

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