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.
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
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