What are the local extrema, if any, of #f (x) =2ln(x^2+3)x#?
We have:
We can identify the critical points by finding where the first derivative equals zero:
so the critical points are:
Now we know that a second order polynomial with positive leading coefficient is positive outside the interval comprised between the roots and negative in the interval between the roots, so that:
graph{2ln(x^2+3) x [1.42, 8.58, 0.08, 4.92]}
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To find the local extrema of ( f(x) = 2\ln(x^2 + 3)  x ), we need to first find the critical points by taking the derivative of ( f(x) ) and setting it equal to zero. Then, we'll determine the nature of these critical points using the second derivative test.

Find the derivative of ( f(x) ): [ f'(x) = \frac{4x}{x^2 + 3}  1 ]

Set ( f'(x) = 0 ) to find critical points: [ \frac{4x}{x^2 + 3}  1 = 0 ] [ 4x = x^2 + 3 ] [ x^2  4x + 3 = 0 ] [ (x  1)(x  3) = 0 ]
So, critical points are ( x = 1 ) and ( x = 3 ).

Find the second derivative of ( f(x) ): [ f''(x) = \frac{4(x^2 + 3)  4x(2x)}{(x^2 + 3)^2} ] [ f''(x) = \frac{4x^2 + 12  8x^2}{(x^2 + 3)^2} ] [ f''(x) = \frac{4x^2 + 12}{(x^2 + 3)^2} ]

Evaluate ( f''(1) ) and ( f''(3) ) to determine the nature of critical points: [ f''(1) = \frac{4(1)^2 + 12}{(1^2 + 3)^2} = \frac{8}{16} = \frac{1}{2} > 0 ] [ f''(3) = \frac{4(3)^2 + 12}{(3^2 + 3)^2} = \frac{36 + 12}{36} = \frac{1}{3} < 0 ]
Since ( f''(1) > 0 ), ( x = 1 ) corresponds to a local minimum. Since ( f''(3) < 0 ), ( x = 3 ) corresponds to a local maximum.
Thus, the local extrema of ( f(x) ) are:
 Local minimum at ( x = 1 )
 Local maximum at ( x = 3 )
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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.
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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