If #f(x)=x(x^2+1) #, what are the points of inflection, concavity and critical points?
Inflection points are where the slope (1st derivative) changes sign (i.e. =0). This is also called a 'critical point'. The curve is concave 'up' when the 2nd derivative is positive, and 'down' when it is negative.
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To find the points of inflection, concavity, and critical points of the function ( f(x) = x(x^2+1) ):

Critical Points: Set the derivative of ( f(x) ) equal to zero and solve for ( x ) to find critical points. ( f'(x) = 0 ) implies ( 3x^2 + 1 = 0 ) This has no real solutions, so there are no critical points.

Concavity: To determine concavity, find the second derivative of ( f(x) ). ( f''(x) = 6x ) Since ( f''(x) ) is positive for all real ( x ), the function is concave up everywhere.

Points of Inflection: Points of inflection occur where the concavity changes. Since the function is always concave up, there are no points of inflection.
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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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