English

Using Rolle'S Theorem, Find Points on the Curve Y = 16 − X2, X ∈ [−1, 1], Where Tangent is Parallel to X-axis. - Mathematics

Advertisements
Advertisements

Question

Using Rolle's theorem, find points on the curve y = 16 − x2x ∈ [−1, 1], where tangent is parallel to x-axis.

Sum
Advertisements

Solution

The equation of the curve is

\[y = 16 - x^2\]  ...(1)

Let P\[\left( x_1 , y_1 \right)\] be a point on it where the tangent is parallel to the x-axis .

Then,
\[\left( \frac{dy}{dx} \right)_P = 0\]   ...(2)
Differentiating (1) with respect to x, we get

\[\frac{dy}{dx} = - 2x\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_P = - 2 x_1 \]

\[ \Rightarrow - 2 x_1 = 0 \left( \text { from } \left( 2 \right) \right)\]

\[ \Rightarrow x_1 = 0\]

\[P\left( x_1 , y_1 \right)\]  lies on the curve\[y = 16 - x^2\] .
\[\therefore\] \[y_1 = 16 - {x_1}^2\]
When \[x_1 = 0\] ,
\[y_1 = 16\]

Hence,\[\left( 0, 16 \right)\] is the required point .

shaalaa.com
  Is there an error in this question or solution?
Chapter 15: Mean Value Theorems - Exercise 15.1 [Page 9]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 15 Mean Value Theorems
Exercise 15.1 | Q 7 | Page 9

RELATED QUESTIONS

f (x) = sin \[\frac{1}{x}\] for −1 ≤ x ≤ 1 Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 4x + 3 on [1, 3] ?


Verify Rolle's theorem for the following function on the indicated interval   f (x) = x(x − 4)2 on the interval [0, 4] ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [−π/4, π/4] ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x on [0, π] ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = ecos x on [−π/2, π/2] ?


Verify Rolle's theorem for the following function on the indicated interval f (x) = \[{e^{1 - x}}^2\] on [−1, 1] ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = 2 sin x + sin 2x on [0, π] ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?


Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?


If f : [−5, 5] → is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?


Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?


Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x2 − 1 on [2, 3] ?


Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = 2x2 − 3x + 1 on [1, 3] ?


Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x2 − 2x + 4 on [1, 5] ?


Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore  f(x) = tan1 x on [0, 1] ?


Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x2 + x − 1 on [0, 4] ?


Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = sin x − sin 2x − x on [0, π] ?


Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?


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


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) ?


Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?


State Lagrange's mean value theorem ?


Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?


If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].

 


For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is 

 


If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]

 


The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is


Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π


An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when θ = `pi/6`


If f(x) = `1/(4x^2 + 2x + 1)`, then its maximum value is ______.


Minimum value of f if f(x) = sinx in `[(-pi)/2, pi/2]` is ______.


Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`


The minimum value of `1/x log x` in the interval `[2, oo]` is


The function f(x) = [x], where [x] =greater integer of x, is


Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :- 


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×