How do you find the local extrema for #f(x) = x - ln(x)# on [0.1,4]?

Answer 1

Local Minima #=f(1)=1.#

Fun. #f# can not have Local Maxima.

Given fun. #f(x)=x-lnx, x in [0.1,4].#
We recall that for local extrema, i.e., maxima/minima, # (i) f'(x)=0, (ii) f''(x)<0# for maxima, &, #f''(x)>0# for minima.
Now, #f'(x)=0 rArr 1-1/x=0 rArr x=1 in [0.1,4]#
#f'(x)=1-1/x rArr f''(x)=0-(-1/x^2)=1/x^2 rArr f''(1)=1>0.#
Therefore, f has a local minima at #x=1#, and it is #f(1)=1-ln1=1-0=1.#
Since, #f''(x)=1/x^2>0, AA x in [0.1,4]#, f can not have any local maxima.
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Answer 2
To find the local extrema for \( f(x) = x - \ln(x) \) on the interval \( [0.1, 4] \), follow these steps: 1. Find the derivative of \( f(x) \) with respect to \( x \). \( f'(x) = 1 - \frac{1}{x} \) 2. Set the derivative equal to zero and solve for \( x \). \( 1 - \frac{1}{x} = 0 \) \( 1 = \frac{1}{x} \) \( x = 1 \) 3. Check the sign of the derivative around the critical point \( x = 1 \) to determine the nature of the extrema. - For \( x < 1 \): \( f'(x) > 0 \), so the function is increasing. - For \( x > 1 \): \( f'(x) < 0 \), so the function is decreasing. 4. Therefore, \( f(x) \) has a local maximum at \( x = 1 \). 5. Check the function values at the endpoints of the interval \( [0.1, 4] \): \( f(0.1) = 0.1 - \ln(0.1) \) \( f(4) = 4 - \ln(4) \) 6. Compare the function values at the critical point, endpoints, and any other critical points within the interval to determine the global extrema. 7. In this case, \( f(0.1) \) and \( f(4) \) should be checked against \( f(1) \). 8. Evaluate \( f(0.1) \) and \( f(4) \) to determine the global extrema.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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