# What are the points of inflection, if any, of #f(x)=1/(x^2-4x+5) #?

Rewrite the function as:

Then differentiate:

Then differentiate again. But now, we're going to have to use chain rule:

Rewrite:

Use common denominators and merge the fractions:

Use quadratic formula:

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To find the points of inflection of ( f(x) = \frac{1}{x^2 - 4x + 5} ), we first find the second derivative and set it equal to zero to locate any inflection points.

The first derivative is ( f'(x) = -\frac{2x - 4}{(x^2 - 4x + 5)^2} ).

The second derivative is ( f''(x) = \frac{2(3x^2 - 8x + 13)}{(x^2 - 4x + 5)^3} ).

Setting ( f''(x) ) equal to zero and solving for ( x ) yields no real solutions.

Thus, there are no points of inflection for the function ( f(x) = \frac{1}{x^2 - 4x + 5} ).

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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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