NCERT Class 12-Mathematics: Chapter – 5 Continuity and Differentiability Part 23 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Question 71:

Find the points on the curve in , where the tangent is parallel to .

Answer:

Given: Equation of curve,

Firstly, we differentiate the above equation with respect to , we get

Given tangent to the curve is parallel to the axis

This means, Slope of tangent Slope of axis

Put in , we have

Hence, the tangent to the curve is parallel to the x – axis at

Question 72:

Using Rolle՚s theorem, find the point on the curve , where the tangent is parallel to .

Answer:

Given:

Now, we have to show that verify the Rolle՚s Theorem

First of all, Conditions of Rolle՚s theorem are:

(a) is continuous at

(b) is derivable at

(c)

If all three conditions are satisfied then there exist some in such that

Condition 1:

On expanding

Since, is a polynomial and we know that, every polynomial function is continuous for all

is continuous at

Hence, condition is satisfied.

Condition 2:

is differentiable at

Hence, condition 2 is satisfied.

Condition 3:

when

when

Hence, condition 3 is also satisfied.

Now, there is atleast one value of

Given tangent to the curve is parallel to the axis

This means, Slope of tangent Slope of axis

Put in , we have

Hence, the tangent to the curve is parallel to the axis at