Explanation Given: The value of f ( 0 ) is 2 and f ′ ( x ) ≤ 3 . Concept used: The mean value theorem: The mean value theorem states that if the function f is continuous on the interval [ a , b ] and differentiable on the interval ( a , b ) then a number c exists in the interval such that the slope at c can be expressed as follows: f ′ ( c ) = f ( b ) − f ( a ) b − a ...... (1) Proof: The mean value theorem can be applied on the interval [ 0 , 4 ] . Substitute 4 for b , 0 for a and x for c in equation (1) as follows: f ′ ( x ) = f ( 4 ) − f ( 0 ) 4 − 0 f ′ ( x ) = f ( 4 ) − f ( 0 ) 4 It is given that f ′ ( x ) ≤ 3 then the above equation can be written as follows: f ( 4 ) − f ( 0 ) 4 ≤ 3 Substitute 2 for f ( 0 ) in the above equation as follows: f ( 4 ) − 2 4 ≤ 3 f ( 4 ) − 2 ≤ 3 ⋅ 4 f ( 4 ) − 2 ≤ 12 f ( 4 ) ≤ 14 For x > 0 , the mean value theorem is applied on the interval [ 0 , x ]

ACHIEVE FOR CALCULUS 1 TERM >CSI CUSTOM<

4th Edition

ISBN: 9781319411657

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 4.3, Problem 69E

To determine

To show:

That f(4)≤14 for [0,4] if f(0)=2 and f′(x)≤3 for x>0 and also prove that f(x)≤2+3x for all x>0.

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Chapter 4.3, Problem 68E

Chapter 4.3, Problem 70E

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Chapter 4 Solutions

ACHIEVE FOR CALCULUS 1 TERM >CSI CUSTOM<

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That f ( 4 ) ≤ 14 for [ 0 , 4 ] if f ( 0 ) = 2 and f ′ ( x ) ≤ 3 for x > 0 and also prove that f ( x ) ≤ 2 + 3 x for all x > 0 .

Solution Summary: The author explains the mean value theorem, which states that if the function f is continuous and differentiable on the interval, then the slope can be expressed as follows.

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To show:

That f(4)≤14 for [0,4] if f(0)=2 and f′(x)≤3 for x>0 and also prove that f(x)≤2+3x for all x>0.

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Chapter 4 Solutions

That f ( 4 ) ≤ 14 for [ 0 , 4 ] if f ( 0 ) = 2 and f ′ ( x ) ≤ 3 for x &gt; 0 and also prove that f ( x ) ≤ 2 + 3 x for all x &gt; 0 .

Solution Summary: The author explains the mean value theorem, which states that if the function f is continuous and differentiable on the interval, then the slope can be expressed as follows.

Concept explainers

Videos

To show: That f(4)≤14 for [0,4] if f(0)=2 and f′(x)≤3 for x>0 and also prove that f(x)≤2+3x for all x>0.

Want to see the full answer?

Chapter 4 Solutions

That f ( 4 ) ≤ 14 for [ 0 , 4 ] if f ( 0 ) = 2 and f ′ ( x ) ≤ 3 for x > 0 and also prove that f ( x ) ≤ 2 + 3 x for all x > 0 .

To show:

That f(4)≤14 for [0,4] if f(0)=2 and f′(x)≤3 for x>0 and also prove that f(x)≤2+3x for all x>0.