Application of Derivatives

Question

At what point will the tangent to the curve y = 2x³ – 15x² + 36x – 21 be parallel to x-axis ? Also, find the equations of tangents to the curve at those points.

Solution

The equation of curve is
y = 2x³ – 15x² + 36x – 21
Differentiating both sides w.r.t.x, we get,
$dy over dx space equals space 6 straight x squared minus 30 straight x plus 36$
For the points on the curve, where tangents are parallel to x-axis
$dy over dx space equals space 0 space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space 6 straight x squared minus 30 straight x plus 36 space equals space 0 space space space rightwards double arrow space space space space straight x squared minus 5 straight x plus 6 space equals space 0$
$therefore space space space space space space left parenthesis straight x minus 2 right parenthesis thin space left parenthesis straight x minus 3 right parenthesis space equals space 0 space space space space space space space space space rightwards double arrow space space space straight x space equals space 2 comma space space 3 therefore space space space space space from space left parenthesis 1 right parenthesis comma space space space straight y space equals space 2 left parenthesis 2 right parenthesis cubed space minus 15 space left parenthesis 2 right parenthesis squared space plus space 36 left parenthesis 2 right parenthesis space minus 21 comma space space space 2 left parenthesis 3 right parenthesis squared space minus space 15 left parenthesis 3 right parenthesis squared space plus 36 left parenthesis 3 right parenthesis minus 21 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 16 minus 60 plus 72 minus 21 comma space space space space 54 minus 135 plus 108 minus 21 space equals space 7 comma space 6 therefore space space space space space space points space are space left parenthesis 2 comma space 7 right parenthesis comma space left parenthesis 3 comma space 6 right parenthesis The space equation space of space tangent space at space left parenthesis 2 comma space 7 right parenthesis space parallel space to space straight x minus axis space is space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 7 space equals space 0 space left parenthesis straight x minus 2 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space space straight m space equals space 0 close square brackets or space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 7 space equals space 0 The space equation space of space tangent space at space left parenthesis 3 comma space 6 right parenthesis space parallel space to space straight x minus axis space is space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 6 space equals space 0 left parenthesis straight x minus 3 right parenthesis space space space space or space space space straight y minus 6 space equals space 0$

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