Home > Calculus > Determining Points of Inflection for a Function

What are the points of inflection of #f(x)=8x - 4 lnx #?

Answer 1

The function has no points of inflection.

Find the points when #f''(x)# switches sign (when #f# switches concavity).

#f'(x)=8-4/x#

#f''(x)=4/x^2#

#f''(x)# is always positive (or undefined when #x=0#), thus the function never switches concavity and has no points of inflection.

graph{8x-4lnx [-9.45, 56.4, 91.34, -40.3]}

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Answer 2

William Abdalla

To find the points of inflection of ( f(x) = 8x - 4 \ln(x) ), you need to determine where the second derivative changes sign.

First, find the second derivative: [ f'(x) = 8 - \frac{4}{x} ] [ f''(x) = \frac{4}{x^2} ]

Next, set ( f''(x) = 0 ) to find any possible points of inflection: [ \frac{4}{x^2} = 0 ] [ x^2 \neq 0 ]

Since the second derivative is never equal to zero, there are no points of inflection for this function.

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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.