Proof of LaGrange's Mean Value theorem

Q.

Q. If the line y = x touches the curve y = x² + bx + c at a point (1, 1) then

(a) b = 1, c = 2
(b) b = −1, c = 1
(c) b = 2, c = 1
(d) b = −2, c = 1

Q. A point on the curve $y=x^2+x$, where the tangent is parallel to the chord joining the points (0, 0) and (1, 2) is .

1. $\left(\frac{1}{2},\frac{3}{4}\right)$
2. $\left(\frac{1}{4},\frac{3}{2}\right)$
3. $\left(\frac{3}{4},\frac{3}{2}\right)$
4. $\left(\frac{1}{2},\frac{5}{4}\right)$

Q. The point on the curve y² = x where tangent makes 45° angle with x-axis is

(a) (1/2, 1/4)
(b) (1/4, 1/2)
(c) (4, 2)
(d) (1, 1)

Q.

Find the slope of the tangent to curve y = x³ − x + 1 at the point whose x-coordinate is 2.

Q. Find the points on the curve y = x³ − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2).

Q. The point at the curve y = 12x − x² where the slope of the tangent is zero will be

(a) (0, 0)
(b) (2, 16)
(c) (3, 9)
(d) none of these

Q.

Show that the tangents to the curve y = 7x³ + 11 at the points where x = 2 and x = −2 are parallel.

Q. Find the points on the curve x² + y² = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7.

Q. 75. Find area bounded by tangents at the intersection of curve with x-axis and curve itself where curve is y = (x-1).(3-x)

Q. Find the point on the curve y = x² where the slope of the tangent is equal to the x-coordinate of the point.

Q. When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is

(a) e^1/1−e

(b) e^{(e−1)(2e−1)}

(c) $e^{\frac{2e-1}{e-1}}$

(d) $\frac{e-1}{e}$

Q. Find a point on the curve y = x² + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2).

Q. At what points on the curve y = 2x² − x + 1 is the tangent parallel to the line y = 3x + 4?

Q.

The slope of the tangent to the curveat the point (2, −1) is

(A) (B) (C) (D)

Q. Find a point on the curve y = x³ + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28).

Q. If the tangent to a curve at a point (x, y) is equally inclined to the coordinates axes then write the value of $\frac{dy}{dx}$.

Q. Find a point on the parabola y = (x − 3)², where the tangent is parallel to the chord joining (3, 0) and (4, 1).

Q.

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).

Q. If the tangent to the curve x = a t², y = 2 at is perpendicular to x-axis, then its point of contact is

(a) (a, a)
(b) (0, a)
(c) (0, 0)
(d) (a, 0)

Q. Find the points on the curve 2a²y = x³ − 3ax² where the tangent is parallel to x-axis.

Q. Find the points on the curve y = 3x² − 9x + 8 at which the tangents are equally inclined with the axes.

Q. Find the equation of the tangent to the curve which is parallel to the 4x − 2y + 5 = 0.

Q.

Find the slope of the tangent to curve y = x³ − x + 1 at the point whose x-coordinate is 2.

Q. The slope of the tangent to the curve x = 3t² + 1, y = t³ −1 at x = 1 is

(a) 1/2
(b) 0
(c) −2
(d) ∞

Q. A point on the curve $y=x^2+x$, where the tangent is parallel to the chord joining the points (0, 0) and (1, 2) is .

1. $\left(\frac{1}{2},\frac{3}{4}\right)$
2. $\left(\frac{1}{4},\frac{3}{2}\right)$
3. $\left(\frac{3}{4},\frac{3}{2}\right)$
4. $\left(\frac{1}{2},\frac{5}{4}\right)$

Q. Ordinate of the point on the curve y=x^x where tangent drawn to the curve is parallel to x axis is a)e^(1/e) b) 1/e c) e^(-1/e) d) 1/(e^e)

Q. 18. Find area bounded by tangents at the intersection of curve with x-axis and curve itself where curve is y = (x-1).(3-x)