Find points at which the tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis.
It is given that the curve y = x3 – 3x2 – 9x + 7
We know that the tangent is parallel to the x-axis if the slope of the tangent is zero.
Therefore, 3x2 -6x-9 = 0
⇒ x2 -2x-3 = 0
⇒ (x-3)(x+1) =0
⇒ x =3 and x = -1
When x = 3, y = (3)3-3(3)2-9(3) +7 = 27 – 27 -27 + 7 = -20
When x = -1, y = (-1)3-3(-1)2-9(-1) +7 = -1 – 3 + 9 + 7 = 12
Therefore, the points at which the tangent is parallel to the x- axis are (3, -20) and (-1, 12).
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