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What are the points of inflection of #f(x)= 10(x-5)^3+2#?

Answer 1

#(5,2)#

Points of inflection, where concavity changes, occur when the second derivative of the function is zero, that is, when #f''(x)=0#.

By using the power rule, we can find the second derivative:

#therefore f'(x)=30(x-5)^2#

#therefore f''(x)=60(x-5)#.

#therefore f''(x)=0 iff x=5#.

But #f(5)=10(5-5)^3+2 = 2#.

Hence the inflection point is #(5,2)#.

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Answer 2

Evelyn Agard

To find the points of inflection of ( f(x) = 10(x-5)^3 + 2 ), we first compute its second derivative, ( f''(x) ), then solve for values of ( x ) where ( f''(x) = 0 ) or does not exist.

First derivative: ( f'(x) = 30(x-5)^2 ) Second derivative: ( f''(x) = 60(x-5) )

Now, setting ( f''(x) = 0 ), we get: [ 60(x-5) = 0 ] [ x - 5 = 0 ] [ x = 5 ]

Thus, the point of inflection is at ( x = 5 ).

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Answer from HIX Tutor

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.