Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean

Question

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers
that satisfy the conclusion of the Mean Value Theorem.

 

f(x) = ln(x)  [1, 4]

 

 

ANSWER: 3/ ln(4)

#11, p288

Answer 1

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 states that the slope of the secant line connecting the points (x_1, f(x₁)), (x₂, f(x₂)) equals the slope of the tangent line at some c in the open interval (x₁, x₂)

 

The slope of the secant line, say m = ln(4) - ln(1) / (4-1) = ln(4) / 3

 

f'(x) = 1/x

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

1/x = ln(4) / 3

x = 3 / ln(4)

 Since 3 / ln(4) ~ 2.16, this value of x does indeed fall in the open interval (1,4) and so satisfies the conclusion of the mean value theorem. Therefore the function satisfies the conclusion of the mean value theorem on [1, 4] with c = 3 / ln 4