Advertisements
Advertisements
Question
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
Options
(2, –2), (–2, –34)
(2, 34), (–2, 0)
(0, 34), (–2, 0)
(2, 2), (–2, 34)
Advertisements
Solution
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are (2, 2), (–2, 34).
Explanation:
Given that y = x3 – 12x + 18
Differentiating both sides w.r.t. x, we have
⇒ `"dy"/"dx"` = 3x2 – 12
Since the tangents are parallel to x-axis, then `"dy"/"dx"` = 0
∴ 3x2 – 12 = 0
⇒ x = ± 2
∴ `y_(x = 2)` = (2)3 – 12(2) + 18
= 8 – 24 + 18
= 2
`y_(x = -2)` = (– 2)3 – 12 (– 2) + 18
= – 8 + 24 + 18
= 34
∴ Points are (2, 2) and (– 2, 34).
APPEARS IN
RELATED QUESTIONS
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
Find a point on the curve y = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Find the equations of the tangent and normal to the given curves at the indicated points:
x = cos t, y = sin t at t = `pi/4`
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.
Find the equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
(A) 1
(B) 2
(C) 3
(D) 1/2
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
Find the angle of intersection of the curves y2 = x and x2 = y.
The abscissa of the point on the curve 3y = 6x – 5x3, the normal at which passes through origin is ______.
The equation of the normal to the curve y = sinx at (0, 0) is ______.
The point on the curve y2 = x, where the tangent makes an angle of `pi/4` with x-axis is ______.
Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
