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Dec 21, 2020 · The number line in Figure \(\PageIndex{5}\) illustrates the process of determining concavity; Figure \(\PageIndex{6}\) shows a graph of \(f\) and \(f''\), confirming our results. Notice how \(f\) is concave down precisely when \(f''(x)<0\) and concave up when \(f''(x)>0\).
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.
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
How to find the concavity of a function. There are a number of ways to determine the concavity of a function. If given a graph of f(x) or f'(x), determining concavity is relatively simple. Otherwise, the most reliable way to determine concavity is to use the second derivative of the function; the steps for doing so as well as an example are ...
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-way conditional logic statement! At the point (0, 0) on the graph of f ( x ) = x 4 , both.
Apr 24, 2022 · The second derivative tells us if a function is concave up or concave down. 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 ...
Find the intervals on which f is increasing/decreasing and the local maxima and minima values for f(x) = x 2sin(x); 0 <x<3ˇ. We begin by taking the derivative of f:
