What are the points of inflection, if any, of #f(x)=x^4x^3+6 #?
To take the second derivative of a function, just derive it twice.
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See explanation.
The zero or zeros of the second derivative can be used to calculate the point of inflection. Here we have:
graph{12x^26x [1.538, 1.54, 3.0777, 3.08]}
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To find the points of inflection for ( f(x) = x^4  x^3 + 6 ):

Find the second derivative ( f''(x) ) by differentiating ( f'(x) ), which is the derivative of ( f(x) ), with respect to ( x ). [ f'(x) = 4x^3  3x^2 ] [ f''(x) = 12x^2  6x ]

Set the second derivative equal to zero and solve for ( x ) to find any possible points of inflection. [ 12x^2  6x = 0 ] [ 6x(2x  1) = 0 ] [ x = 0 \quad \text{or} \quad x = \frac{1}{2} ]

Test the concavity of the function around the points obtained in step 2 using the second derivative test:
 If ( f''(x) > 0 ), the function is concave up.
 If ( f''(x) < 0 ), the function is concave down.

Evaluate ( f''(x) ) at ( x = 0 ) and ( x = \frac{1}{2} ):
 At ( x = 0 ), ( f''(0) = 0 ), so the test is inconclusive.
 At ( x = \frac{1}{2} ), ( f''\left(\frac{1}{2}\right) = 3 > 0 ), indicating the function is concave up.

Therefore, the point ( \left(\frac{1}{2}, f\left(\frac{1}{2}\right)\right) ) is a point of inflection for the function ( f(x) = x^4  x^3 + 6 ).
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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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