Chapter 3: Q. 14 (page 248)

A function that satisfies the hypothesis, and therefore the
conclusion, of the Mean Value Theorem.

Short Answer

Expert verified

At point $(1, 4)$ the function satisfied the conditions of Rolle's theorem .

Step by step solution

Step 1. Given information .

A function that satisfies the hypothesis, of the Rolle's theorem .

Step 2. ln (x) is indeed continuous on [1,4] and differentiable on (1,4) therefore it satisfies the hypothesis of the mean value theorem.

The Mean value theorem stated that the secant line connecting the points

(x_{1} f (x_{1})), (x_{2} f (x_{2}))

equals the slope of the tangent line at some c in the open interval

(x_{1}, x_{2})

The slope of the secant line role="math" localid="1648627193201" $m = \frac{I n (4) - I n (1)}{4 - 1} = \frac{I n (4)}{3}$ .

$f' (x) = \frac{1}{x}$

Setting the derivative equal to the slope of secant and solving for x .

$\frac{1}{x} = \frac{I n (4)}{3} x = \frac{3}{I n (4)}$

Since this value $\frac{3}{I n (4)}$ of x indeed fall in open interval $(1, 4)$ so this satisfied the condition of Mean value theorem .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

A function that satisfies the hypothesis, and therefore the
conclusion, of the Mean Value Theorem.

Short Answer

Step by step solution

Step 1. Given information .

Step 2. ln (x) is indeed continuous on [1,4] and differentiable on (1,4) therefore it satisfies the hypothesis of the mean value theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Mechanics Maths

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

A function that satisfies the hypothesis, and therefore theconclusion, of the Mean Value Theorem.

Short Answer

Step by step solution

Step 1. Given information .

Step 2. ln (x) is indeed continuous on [1,4] and differentiable on (1,4) therefore it satisfies the hypothesis of the mean value theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Mechanics Maths

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

A function that satisfies the hypothesis, and therefore the
conclusion, of the Mean Value Theorem.