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Review your knowledge of concavity of functions and how we use differential calculus to analyze it.
- You have two functions f and g, where f''>0 and g''>0. If you take the second derivative of f+g, you get f''+g'', which is positive. So their sum i...
- In order for 𝑓(𝑥) to be concave up, in some interval, 𝑓 ''(𝑥) has to be greater than or equal to 0 (i.e. non-negative) for all 𝑥 in that inter...
- In general, no. Points where f"(x)= 0 are specifically called inflection points. But in the example provided, -2 is also a critical point (Observe...
- Product rule: d/dx[f(x)*g(x)] = f'(x)g(x) + f(x)g'(x) f(x) = (x-1)³ and g(x) = x
State the first derivative test for critical points. Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph. Explain the concavity test for a function over an open interval.
Dec 21, 2020 · The important \(x\)-values at which concavity might switch are \(x=-1\), \(x=0\) and \(x=1\), which split the number line into four intervals as shown in Figure \(\PageIndex{7}\). We determine the concavity on each.
If \( f''(x) \) is positive on an interval, the graph of \( y=f(x) \) is concave up on that interval. We can say that \(f\) is increasing (or decreasing) at an increasing rate. If \( f''(x) \) is negative on an interval, the graph of \( y=f(x) \) is concave down on that interval.
Apr 28, 2020 · The necessary and sufficient condition for a bivariate and twice continuously differentiable function to be concave is that the Hessian matrix be negative semi definite, i.e.: ⎡⎣⎢ ∂2f ∂x2 ∂2f ∂x∂y ∂2f ∂x∂y ∂2f ∂y2 ⎤⎦⎥ ⪯ 0 [∂ 2 f ∂ x 2 ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ x ∂ y ∂ 2 f ∂ y 2] ⪯ 0. which is originated from the Taylor expansion of f(x, y) f (x, y).
How do you describe the concavity of the graph and find the points of inflection (if any) for #f(x) = x^3 - 3x + 2#?
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In this explainer, we will learn how to determine the concavity of a function as well as its inflection points using its second derivative. Before you start with this explainer, you should be confident finding the first and second derivatives of functions using the standard rules for differentiation.
