Finding Concavity and Points of Inflection Graphically
Read this topicExplanations 6
- Concavity is a topic that you'll delve into deeper in calculus. For now, here's a simple understanding of what it is. Concavity and Concave Functions A function is concave (or c…
By Alex Federspiel, Ruiran Xun, Joshua Siktar
- Before Diving In Now that we've covered extrema, let's move on to a more complex topic. Before diving into this lesson, you may want to review how to find the derivative of a fu…
By Joshua Siktar, Anusha Rahman
- In this explanation we're going to talk about concavity and convexity from an algebraic standpoint, with inequalities. A function fff is convex if for all xxx and yyy in the fun…
By Joshua Siktar, Victor Wang, Gabi Slizewska
- In this explanation I'm going to talk about concavity and convexity using mostly graphs. "Concavity" and "convexity" are both often used as qualitative descriptions of the shape…
By Joshua Siktar, Anusha Rahman
- (Video) Finding Concavity and Points of Inflection Graphically by Anusha Rahman In this lesson by Anusha Rahman, you will learn how to find points of inflection and concavity gr…
By Anusha Rahman
- Points of Inflection Points of inflection are the points at which the graph goes from concave to convex, or vice versa. This relates to functions increasing and decreasing.
By Anusha Rahman
Problems 26
- Let f(x)=x2+x+1f(x) = x^2+x+1f(x)=x2+x+1. What is the minimum possible value of f(1−u)+f(1+u)f(1-u) + f(1+u)f(1−u)+f(1+u)?
- Above is the graph of the function f(x)=2xf(x)=2^xf(x)=2x. Is it concave or convex?
- Above is the graph of the function f(x)=2sin(x)f(x)=2\sin(x)f(x)=2sin(x). Select the intervals on which the function is concave and convex.
- Above is the graph of the function f(x)=tan(x)f(x)=\tan(x)f(x)=tan(x). Select the intervals on which the function is concave and convex.
- Above is the graph of the function f(x)=ln(x)f(x)=\ln(x)f(x)=ln(x). Is it concave or convex?
- True or False\text{True or False}True or False === A function can be both concave and convex on some domain.
- How many times is the graph above concave and convex? Select all answers that apply.
- The function f(x)=ln(ax)f(x)=\ln(ax)f(x)=ln(ax) is concave for what values of aaa?
- Does the table above show a function that is convex or concave?
- If f(x)f(x)f(x) is a convex (concave up) function, which of the functions listed below must also be convex? Select all that apply.
- Which of the following inequalities are true for all real numbers xxx and y?y?y?
- Consider the following equation-inequality system: x2+y=0x^2 + y = 0x2+y=0 (x+y)2+4ex2+y2≤2(x2+ex+y2+ey)(x + y)^2 + 4e^{\frac{x}{2} + \frac{y}{2}} \leq 2(x^2 + e^x + y^2 + e^y)(…
- Which of the following inequalities are true for all positive x1x_1x1 and x2x_2x2 because of the algebraic definition of a concave function?
- On which interval is the following inequality true for all values of xxx and yyy in that interval? Give the largest possible interval. (13x+23y)3≤13x3+23y3.\left(\frac{1}{3}x + …
- On which interval is the following inequality true for all values of xxx and yyy in that interval? Give the largest possible interval. (13x+23y)3333≤13x3333+23y3333.\left(\frac{…
- Above is the graph of the function f(x)=2xf(x)=2^xf(x)=2x. On which interval is the function convex?
- Does the table above show a function that has an increasing or decreasing slope?
- If f(x)f(x)f(x) is a convex function, which of the following must also be convex?
- Above is the graph of the function f(x)=tan(x)f(x)=\tan(x)f(x)=tan(x). Select the inflection points of the graph that are displayed.
- Let fff and ggg be convex functions each with domain (−∞,∞).(-\infty, \infty).(−∞,∞). Under what condition is f+gf + gf+g also a convex function with domain (−∞,∞)?(-\infty, \in…
- Suppose fff is a convex function and ggg is a concave function. Which of the following guarantees that f−gf - gf−g is also convex?
- If a function is always concave up, how many inflection points does it have?
- If a function is always concave down, how many inflection points does it have?
- Suppose a function fff has domain R,\mathbb{R},R, being concave up on (−∞,0)(-\infty, 0)(−∞,0) and concave down on (0,∞).(0, \infty).(0,∞). Which of the following are inflection…
- Suppose a function fff has domain R,\mathbb{R},R, being concave down on (−∞,0)(-\infty, 0)(−∞,0) and concave up on (0,∞).(0, \infty).(0,∞). Which of the following are inflection…
- Let λ∈(0,1).\lambda \in (0, 1).λ∈(0,1). Find the value of limx→∞ln(xλ(x+1)λ−1)λx+(1−λ)(x+1).\lim_{x \rightarrow \infty}\frac{\ln\left(\frac{x^{\lambda}}{(x + 1)^{\lambda - 1}}…