How do you find the relative extrema for #f(x) = x  log_4x#?
The minimum of
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To find the relative extrema for ( f(x) = x  \log_4(x) ), follow these steps:
 Take the derivative of ( f(x) ) with respect to ( x ).
 Set the derivative equal to zero and solve for ( x ).
 Check the second derivative at each critical point to determine whether it's a relative minimum, maximum, or neither.
Now, let's go through these steps:

The derivative of ( f(x) = x  \log_4(x) ) is: [ f'(x) = 1  \frac{1}{x \ln(4)} ]

Setting ( f'(x) ) equal to zero and solving for ( x ): [ 1  \frac{1}{x \ln(4)} = 0 ] [ 1 = \frac{1}{x \ln(4)} ] [ x \ln(4) = 1 ] [ x = \frac{1}{\ln(4)} ]

To determine the nature of the critical point, we need to check the second derivative: [ f''(x) = \frac{1}{x^2 \ln^2(4)} ] Substituting ( x = \frac{1}{\ln(4)} ) into ( f''(x) ), we get: [ f''\left(\frac{1}{\ln(4)}\right) = \frac{1}{\left(\frac{1}{\ln(4)}\right)^2 \ln^2(4)} = \ln^2(4) ]
Since the second derivative is positive, ( x = \frac{1}{\ln(4)} ) corresponds to a relative minimum.
Therefore, the relative minimum for ( f(x) = x  \log_4(x) ) occurs at ( x = \frac{1}{\ln(4)} ).
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When evaluating a onesided 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 onesided 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 onesided 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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