Tamil Nadu Board of Secondary EducationHSC Arts Class 12th

# Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions: f(x)=x2-2xx+2,x∈[-1,6] - Mathematics

Sum

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:

f(x) = (x^2 - 2x)/(x + 2), x ∈ [-1, 6]

#### Solution

f(– 1) = (1 + 2)/(-1 + 2) = 3

f(6) = (36 - 12)/8 = 24/8 = 3

⇒ f(– 1) = 3 = f(6)

f(x) is continuous on [– 1, 6]

f(x) is differentiable on (– 1, 6)

Now, f'(x ) = ((x + 2)(2x - 2) - (x^2 - 2x)(1))/(x + 2)^2

= (x^2 + 4x - 4)/(x + 2)^2

Since the tangent is parallel to the x-axis.

f'(x) = 0

⇒ x2 + 4x – 4 = 0

⇒ x = - (4 +-  sqrt(16 + 16))/2

x = - (4 +-  4sqrt(2))/2

= - 2 +-  2sqrt(2)

x = - 2 +- 2sqrt(2) ∈ (-1, 6)

Concept: Mean Value Theorem
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#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 12th Mathematics Volume 1 and 2 Answers Guide
Chapter 7 Applications of Differential Calculus
Exercise 7.3 | Q 2. (ii) | Page 21
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