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Definition 1. A function f : S ⊂ Rn → R defined on a convex set S is concave if for any two points x1 x2 ∈ , S and for any λ ∈ [0, 1] we have: λx1 (1 − λ) x2 ≥ λf(x1) (1 − λ)f(x2) + +. is called strictly concave if for any two points x1 , x2 ∈ S and for any λ ∈ (0, 1) we have: λx1 (1 − λ) x2 > λf(x1) (1 − λ)f(x2) + +.
Dec 21, 2020 · If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
A point on a graph where the concavity of the curve changes (from concave down to concave up, or vice versa) is called a point of inflection (Definition 4.14). By implication (think about what separates positive and negative numbers on a number line), if a point (c, f (c)) is a point of inflection, then f ′ (c) = 0. Important: This is a one ...
- Definition 1
- Definition 2
- Fact
- Second Derivative Test
To show that the graphs above do in fact have concavity claimed above here is the graph again (blown up a little to make things clearer). So, as you can see, in the two upper graphs all of the tangent lines sketched in are all below the graph of the function and these are concave up. In the lower two graphs all the tangent lines are above the graph...
Now that we have all the concavity definitions out of the way we need to bring the second derivative into the mix. We did after all start off this section saying we were going to be using the second derivative to get information about the graph. The following fact relates the second derivative of a function to its concavity. The proof of this fact ...
So, what this fact tells us is that the inflection points will be all the points where the second derivative changes sign. We saw in the previous chapter that a function may change signs if it is either zero or does not exist. Note that we were working with the first derivative in the previous section but the fact that a function possibly changing ...
The third part of the second derivative test is important to notice. If the second derivative is zero then the critical point can be anything. Below are the graphs of three functions all of which have a critical point at x=0x=0, the second derivative of all of the functions is zero at x=0x=0and yet all three possibilities are exhibited. The first i...
Dec 21, 2020 · When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum. We have been learning how the first and second derivatives of a function relate information about the graph of that function.
If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
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The graph of \(f(x)=x^{3}\) shows that \(f\) is concave down on \((-\infty,0)\) and concave up on \((0,+\infty)\). Hence, \((0,f(0))=(0,0)\) is an inflection point of the graph of \(f\). (c) In Example 1, we saw that the direction of concavity changes at \(x=1\) .
