# How do you determine whether the function #f (x) = x sqrt(x^2+2x+5)+1 sqrt(x^2+2x+5)# is concave up or concave down and its intervals?

It's concave up for

Now use the Quotient Rule and Chain Rule to find the second derivative:

So

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To determine whether the function ( f(x) = x \sqrt{x^2+2x+5} + \sqrt{x^2+2x+5} ) is concave up or concave down, we first find its second derivative, denoted as ( f''(x) ). Then, we analyze the sign of ( f''(x) ) to identify the concavity of the function and its intervals.

After computing the second derivative, we evaluate it at critical points and any other points of interest to determine where the function changes concavity. The critical points are found by setting the first derivative equal to zero and solving for ( x ).

If ( f''(x) > 0 ) for a particular interval, the function is concave up on that interval. If ( f''(x) < 0 ), the function is concave down on that interval.

In summary, to determine concavity and its intervals:

- Compute the second derivative ( f''(x) ).
- Find critical points by setting the first derivative equal to zero and solving for ( x ).
- Evaluate ( f''(x) ) at critical points and any other points of interest.
- Analyze the sign of ( f''(x) ):
- If ( f''(x) > 0 ), the function is concave up on the interval.
- If ( f''(x) < 0 ), the function is concave down on the interval.

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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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- How do you find the inflection point of the function #f(x) = x^2ln(x)#?
- How do you find all points of inflection given #y=-2sinx#?
- Is #f(x)=1-xe^(-3x)# concave or convex at #x=-2#?

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