Math question on differential calculus
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Math question on differential calculus

[From: ] [author: ] [Date: 11-08-09] [Hit: ]
where a and b are constants, has equation 4x+y=6 at the point where x=1. Find the values of a and b.The tangent at x = 1 is 4x+y = 6 which has a slope = y = -4 and the point (x,y) = (1,The curve is,......
I've been trying to get this question but I just don't understand how I'm supposed to go about finding the answer! Help!

The tangent to the curve y=x^2+ax+b, where a and b are constants, has equation 4x+y=6 at the point where x=1. Find the values of a and b.

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y = x² + ax + b
y' = 2x + a

y(1) = 1+a+b
y'(1) = 2+a

The tangent at x = 1 is 4x+y = 6 which has a slope = y' = -4 and the point (x,y) = (1, 2)

Since the curve and the tangent line share the point of tangency and the slope at that point:
y'(1) = -4 = 2+a ==> a = -6

The curve
y(1) = 2 = 1 + a + b
2 = 1 - 6 + b
b = 7

The curve is, with a = -6 and b = 7
y = x² -6x + 7
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