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If the tangent to the curve, y = x³ + ax − b at the point (1, −5) is perpendicular to the line, −x + y + 4 = 0, then which of the following points lies on the curve?
- (−2, 1)
- (−2, 2)
- (2, −1)
- (2, −2)
Correct answer: (2, −2)
Solution
The slope of the line given is 1, so the slope of the tangent to the curve must be -1 for them to be perpendicular. By finding the derivative of the curve and substituting the point (1, -5), we can determine the values of a and b. After solving for these constants, we can check which of the provided points satisfies the curve equation, confirming that (2, -2) is indeed a solution.
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