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A curve defined by y = ax³ + bx² + cx + 1 is tangent to the x-axis at the point (-2, 0) and crosses the y-axis at a point where the slope of the curve is 3. What is the value of a + b + c?

1. 23/4
2. 25/4
3. 6
4. 3

Correct answer: 23/4

Solution

Using the three conditions y(-2)=0, y'(-2)=0, and y'(0)=3, we find c=3, then solve the system for a and b. Adding a+b+c gives 5/4 +... the conditions yield a=3/4, b=0, c=3, giving a+b+c = 3/4 + 0 + 3 = 15/4. Re-examining: y'(x)=3ax²+2bx+c; y'(0)=c=3. y(-2)=-8a+4b-2c+1=0 => -8a+4b=5. y'(-2)=12a-4b+c=0 => 12a-4b=-3. Adding: 4a=2 => a=1/2. Then 4b=5+8*(1/2)=5+4=9 => b=9/4. a+b+c=1/2+9/4+3=2/4+9/4+12/4=23/4.

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