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 one of the following points lies on the curve?
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Correct option is B)
y=x2+ax−b (1,−5) lies on the curve ⇒−5=1+a−b⇒a−b=−6 .(i) Also, y′=3x2+a y(1,−5)′=3+a (slope of tangent) ∵ this tangent is ⊥ to −x+y+4=0 ⇒(3+a)(1)=−1 ⇒a=−4 .(ii) By (i) and (ii): a=−4, b=2 ∴y=x3−4x−2 (2,−2) lies on this curve.