Mathmatic Help?

Question

I have 2 questions that i am strugaling to answer

1) A curve passes through the origin and touches the x-axis at (4,0)
Given that g(x)= x(cubed)+lx(squared) + mx + n
I have to find the values of l, m and n, but i cant understand how i do it :S.

2) State the co-ordinates of the minimum point on the graph of y=x(squared) - 8x +25

Answer 1

1) The curve passes through the origin (0,0). Therefore, g(0) = 0:

0 = 0³ + l*0² + m*0 + n

Therefore, n=0

Similarly, since the curve passes through (4,0), we have g(4) = 0:

0 = 4³ + l*4² + m*4 = 64 + 16l + 4m

And since the curve touches the x-axis at (4,0), we also have g'(4) = 0:

g'(x) = 3x² + 2lx + m

g'(4) = 0 → 0 = 3*4² + 2l*4 + m = 48 + 8l + m

Solve 64 + 16l + 4m = 0 and 48 + 8l + m = 0 simultaneously for l and m.

2) Complete the square in the expression for y and argue that, since a squared (real) expression is at least zero, the minimum value for y occurs when that squared expression is zero.

This is better than setting y' to zero and solving for x, since doing that only gives you critical points. And even if you use the first or second derivative test on the one critical point, that only establishes that the critical point is a relative minimum, whereas the method I've suggested first shows that it is a global minimum.