MTH101 Assignment 3 Deadline 7 July 2010

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Best Answer – Q1

y = x^2+k\x = x^2 + kx^(-1)

Max/min points are found when the gradient = 0
Find the gradient:

dy/dx = 2x - kx^(-2)
2x - kx^(-2) = 0

Taking LCM

2x^3 - k =0
2x^3 = k

If x=3:

k = 2(3)^3 = 2(27) = 54

k = 54



Q2
f'(x) = 1 - 1/x^2 > or = 0 in [1, +oo]
So, f(x) is increasing.

fmin = f(1) = 1 + 1/1 = 2
x + 1/x > fmin = 2


f'(x) = 1 - 1/x^2 = (x^2 - 1)/x^2.

Then, if f(x) is increasing, then f'(x) > 0 and so:

(x^2 - 1)/x^2 > 0.

Then, since x^2 is positive for all x, we have:

x^2 - 1 > 0
==> x < -1 and x > 1.

Thus, f(x) is increasing for (1, infinity).

For the second part, since f(x) is increasing for x > 1:
x > 1 <==> f(x) > f(1).

Since f(1) = 1 + 1/1 = 2, we see that for x > 1:
f(x) > 1 + 1/1 = 2. ∴

I hope this helps!