Hi my textbook uses the second derivative method to find inflection points. Now since inflection points are when the concavity changes direction, and if the graph of f is concave upward if f' is increasing on that interval and vice versa, isn't it possible to just use the first derivative to find points of inflection?
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How will you find where f' changes from decreasing to increasing?
To find a point of inflection using the first derivative, you need to know where that first derivative reaches a minimum or maximum. For some functions (as first derivatives) that might be easy. (For example, if your original function is sin(x), the first derivative will be cos(x), and you know its maxima will be at x=2nπ and its minima will be at x=(2n+1)π, where n is an integer.) More generally, though, how would you do it?
The first derivative is itself a function. So, it's really just the question: how would you find the maxima and minima of a function? Whatever your answer to that question is, that's the answer to the question of how you'd identify maxima and minima in the first derivative, and therefore identify points of inflection in the original function.
So, your original question is equivalent to the following: 'isn't it possible to just use the original function to find its own maxima and minima?'
I would take a random number from that interval, plug it into the derivative, and if it was positive, it is increasing, and vice versa.
That wouldn't tell you where the maxima or minima are, so you wouldn't find any points of inflection that way, either.
How would you find the maxima and minima of a function f(x)?
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|dw:1354767370733:dw|The first derivative tells you whether the original function is increasing or decreasing. A function can increase and be concave up or concave down. Likewise, it can decrease and be concave up or down. If I take a random number within an interval and plug it into the derivative, and if it is positive it is increasing... but I have no idea if it is concave up or down.
In your OP you made a statement that is very true "the graph of f is concave up if f' is increasing". If f' is positive, it indicates that f is increasing. If f' is increasing, then it is concave up... but being positive does not indicate that f' is increasing... the derivative of f' being positive is the indicator that f' is increasing. And the derivative of f' is f".
Oh I see! Thank You guys!
The second derivative gives the character of the first derivative, thus if f''(x) is positive zero, greater than zero, or less than zero, it gives you the nature of the slope all around the curve without necessarily sketching. at a time it will be good to work the rigorous approach you talking about, but as you mature up, just know what the sh