maxima, minima, and inflection points

[logistic_guy]

Find the critical numbers of \(\displaystyle f(x) = 5 - 2x + x^2\), and determine whether they yield relative maxima, relative minima, or inflection points.

 

[khansaheb]

Find the critical numbers of \(\displaystyle f(x) = 5 - 2x + x^2\), and determine whether they yield relative maxima, relative minima, or inflection points.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem

 

[logistic_guy]

We can find the critical numbers when we set the first and the second derivatives of \(\displaystyle f(x)\) to zero.

\(\displaystyle f'(x) = -2 + 2x\)
\(\displaystyle 0 = -2 + 2x\)

This gives:

\(\displaystyle x = 1\)

The second derivative is:

\(\displaystyle f''(x) = 2\)

Since we cannot set the second derivative to zero, we know that there is no inflection points.

In the next post, we will investigate more to know if \(\displaystyle x = 1\) yields a relative maxima or a relative minima.

 

[logistic_guy]

There are many ways to tell if the critical number yields a relative maxima or a relative minima. One of them is to substitute it in the second derivative.

\(\displaystyle f''(1) = 2\)

Since \(\displaystyle 2 > 0\), \(\displaystyle x = 1\) yields a relative minima.

In fact, it yields an absolute minima. Why?

Because when the second derivative is positive for all \(\displaystyle x\), then the function \(\displaystyle f\) is concave up for all \(\displaystyle x\).