When you're finding the sign of f' and f'' to find a local maximum/minimum, concavity and inflection point where do you substitute the tangent/critical points? In which function? f(x), f'(x) or f''(x)?
First determine the points where f'(x) = 0 Then evaluate f''(x) at those same points to determine inflection, concavity and maximum/minimum.
rather than going through a lengthy explanation, I will use an example to answer your question.
e.g. f(x) = x^3 - 6x^2 + 1
f'(x) = 3x^2 - 12x
set this to zero, to find local max/mins
3x^2 - 12x = 0
3x(x-4) = 0
x = 0 or x = 4, then
y = 1 or y = -31
then (0,1) and (4,-31) are local max/mins, but I don't know at this point whether they are a maximim or minimum
f''(x) = 6x-12
if f''(x) is positive for some x, then the curve is concave upwards at that point, and if f''(x) is negative for some x, then the curve is concave downwards at that x.
so we sub (0,1) into 6x-12 to get -12
so at (0,1) the curve opens downwards, so (0,1) must be a maximum point
similarly at (4,-31), f''(4) = 24-12 = 12, positive, so at (4,-31) the point opens upwards, implying (4,-31) is a minimum point.
notice when I set f''(x) = 0,
6x-12 = 0 I get x = 2, and y = -15
so the point of inflection is (2,-15)
which happens to be the midpoint between the max and min points. This is always true for a general cubic function, and makes an interesting property to prove.
Also notice that f'(x) indicates whether the curve is increasing or decreasing at any x value that you pick.
Make yourself a summary to learn which derivative is used for what.
When determining the sign of f' (the first derivative) and f'' (the second derivative) to find the local maximum/minimum, concavity, and inflection points, you need to substitute the x-values of the critical points into the corresponding derivative functions.
Let's break it down step by step:
1. Find the critical points: To locate the critical points, you need to set the first derivative, f'(x), equal to zero (0) or undefined (if it exists). Solve the resulting equation(s) to find the x-values where the derivative is zero or undefined.
2. Determine the interval boundaries: Identify the x-values that may potentially affect the sign of the derivative and second derivative. These are typically the endpoints of the interval under consideration.
3. Substitute into f'(x) and f''(x): Once you have found the critical points and the interval boundaries, substitute the x-values into the first derivative, f'(x), and the second derivative, f''(x). This substitution will allow you to examine the sign of the derivatives at those specific points.
- To find local maximum or minimum points:
- Evaluate f'(x) at the critical points.
- If f'(x) changes sign from positive to negative (as x goes from left to right), it indicates a local maximum.
- If f'(x) changes sign from negative to positive (as x goes from left to right), it suggests a local minimum.
- To determine concavity and inflection points:
- Calculate f''(x) at both the critical points and other points where f'(x) changes sign or is undefined.
- If f''(x) is positive, the function is concave up.
- If f''(x) is negative, the function is concave down.
- Inflection points occur where the concavity changes (f''(x) changes sign or is undefined).
To summarize, you substitute the critical points obtained from f'(x) into f'(x) itself to check for local maximum/minimum points. For concavity and inflection points, you substitute the critical points into f''(x) along with other relevant points obtained from the sign changes or undefined portions of f'(x).