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## Homework Statement

Show f(x)=(x−2)sinxln(x+2) has f'(x)=0 somewhere on [-1,3]

## The Attempt at a Solution

I tried using Rolle's theorem, but f(-1)≠f(3). Then I tried the mean value theorem, but didn't get 0 either.

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- Thread starter jaqueh
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In summary, you tried using Rolle's theorem, the mean value theorem, and numerically, but you didn't get a solution.

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Show f(x)=(x−2)sinxln(x+2) has f'(x)=0 somewhere on [-1,3]

I tried using Rolle's theorem, but f(-1)≠f(3). Then I tried the mean value theorem, but didn't get 0 either.

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- #2

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Check some obvious values in between -1 and 3. Maybe x=0 and x=2? You are giving up too easily.

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jaqueh said:## Homework Statement

Show f(x)=(x−2)sinxln(x+2) has f'(x)=0 somewhere on [-1,3]

## The Attempt at a Solution

I tried using Rolle's theorem, but f(-1)≠f(3). Then I tried the mean value theorem, but didn't get 0 either.

You can find f'(x) analytically, so maybe you should do that and then apply the intermediate value theorem.

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jaqueh said:## Homework Statement

Show f(x)=(x−2)sinxln(x+2) has f'(x)=0 somewhere on [-1,3]

## The Attempt at a Solution

I tried using Rolle's theorem, but f(-1)≠f(3). Then I tried the mean value theorem, but didn't get 0 either.

Well if it's numerical analysis, why are you using those theorems? Suppose it was some function that you didn't even know what the form was, how could you show the derivative is zero somewhere in that interval? If it was mine, I'd generate "numerically" since this is numerical analysis, a set of equally spaced points in that interval and then compute the value of the function and then I'd inspect the list for the "monotonicity" change, that is, when the numbers are increasing then decreasing or vice-a-versa. However if there is an inflection point, the derivative could still be zero without this change in monotonicity. But we could at least rule out the former case this way. Anyway, lots of tough problems won't give you the luxury of using nice theorems and you'll have to muscle-through the data in this way or another.

I got an idea, why don't you write a short Mathematica program to generate these numbers then pick out the monotonicity change?

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jaqueh said:

What does the interval [0,1] do for you?

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LCKurtz said:What does the interval [0,1] do for you?

sorry i meant for the interval to be [0,2]

then i get f(0)=f(2)=0 => rolle's

Last edited:

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jaqueh said:sorry i meant for the interval to be [0,2]

then i get f(0)=f(2)=0 => rolle's

That's better.

Numerical analysis is the study of algorithms and procedures used to solve mathematical problems with the help of computers or calculators. It involves the use of numerical methods to approximate the solutions of problems that cannot be solved analytically.

Numerical analysis is important in calculus because it allows us to find approximate solutions to problems that cannot be solved exactly. This is especially useful in cases where the functions involved are too complex to be integrated or differentiated analytically.

Some common numerical methods used in calculus include the Newton-Raphson method for finding roots of equations, the Trapezoidal Rule and Simpson's Rule for approximating integrals, and the Euler method for solving differential equations.

The accuracy of results obtained from numerical analysis depends on several factors such as the chosen method, the precision of the computer or calculator used, and the complexity of the problem. In general, the more iterations and refinements used, the more accurate the results will be.

Numerical analysis is limited by the fact that it can only provide approximate solutions, which may not always be accurate. It also relies on the use of computers and calculators, which can introduce rounding errors and other sources of error. Additionally, some problems may require a large number of iterations, making the process time-consuming.

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