# How do you find the intervals in which #f (x) = log_7(1+x^2)# is increasing or decreasing?

A function is increasing where the derivative is positive and negative when it is negative.

So:

graph{log(1+x^2)/ln7 [-10, 10, -5, 5]}

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To find the intervals where ( f(x) = \log_7(1+x^2) ) is increasing or decreasing, follow these steps:

- Take the derivative of ( f(x) ) with respect to ( x ) to find the critical points.
- Set the derivative equal to zero and solve for ( x ) to find the critical points.
- Determine the sign of the derivative in each interval defined by the critical points.
- If the derivative is positive, ( f(x) ) is increasing in that interval. If it's negative, ( f(x) ) is decreasing.

Let's find the derivative of ( f(x) ): [ f'(x) = \frac{d}{dx}\left(\log_7(1+x^2)\right) ] [ f'(x) = \frac{1}{\ln(7)} \cdot \frac{d}{dx}\left(\ln(1+x^2)\right) ] [ f'(x) = \frac{1}{\ln(7)} \cdot \frac{2x}{1+x^2} ]

Now, let's set ( f'(x) ) equal to zero and solve for ( x ): [ \frac{1}{\ln(7)} \cdot \frac{2x}{1+x^2} = 0 ] [ 2x = 0 ] [ x = 0 ]

Now, we have a critical point at ( x = 0 ).

To determine the intervals of increase and decrease, consider the sign of ( f'(x) ) in the intervals defined by the critical point ( x = 0 ).

- When ( x < 0 ), ( f'(x) ) is negative, so ( f(x) ) is decreasing.
- When ( x > 0 ), ( f'(x) ) is positive, so ( f(x) ) is increasing.

Therefore, the function ( f(x) = \log_7(1+x^2) ) is increasing for ( x > 0 ) and decreasing for ( x < 0 ).

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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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