A) \[(30-20)\]and\[(-1,12)\]
B) (3, 20) and (1, 12)
C) \[(1,-10)\]and (2, 6)
D) None of these
Correct Answer: A
Solution :
Tangent to the curve is parallel to the axis is when slope of the tangent is 0. \[\therefore \] Equation of the curve is \[y={{x}^{3}}-3{{x}^{2}}-9x+7\] ?(i) \[\therefore \] \[\frac{dy}{dx}=3{{x}^{2}}-6x-9\] Now, the tangent is parallel to \[x-\]axis, then slope the tangent is zero or we can say that\[\frac{dy}{dx}=0.\] \[\Rightarrow \] \[3{{x}^{2}}-6x-9=0\] \[\Rightarrow \]\[3({{x}^{2}}-2x-3)=0\] \[\Rightarrow \]\[(x-3)(x+1)=0\] \[\Rightarrow \]\[x=3,-1\] When \[x=3,\]then from Eq. (i), we get \[y={{(3)}^{3}}-(3).{{(3)}^{2}}-9.3+7\] \[=27-27-27+7=-20\] When \[x=-1,\]then from Eq. (i), we get \[y={{(-1)}^{3}}-3{{(-1)}^{2}}-9(-1)+7\] \[=-1-3+9+7=12\] Hence, the points at which the tangent is parallel to \[x-\]axis are \[(3,-20)\] and \[(-1,12).\]
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