Search results
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
Dec 21, 2020 · Let \(f(x)=x/(x^2-1)\). Find the inflection points of \(f\) and the intervals on which it is concave up/down. Solution. We need to find \(f'\) and \(f''\). Using the Quotient Rule and simplifying, we find \[f'(x)=\frac{-(1+x^2)}{(x^2-1)^2} \quad \text{and}\quad f''(x) = \frac{2x(x^2+3)}{(x^2-1)^3}.\]
How do you determine the values of x for which the graph of f is concave up and those on which it is concave down for #f(x) = 6(x^3) - 108(x^2) + 13x - 26#? Is the function concave up or down if #f(x)= (lnx)^2#?
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 ...
Investigating concavity enhances our ability to comprehend the dynamics of functions, enabling precise predictions about their increasing, decreasing, or stationary nature in various intervals. Questions. For what values of x is f (x)= -x^3+3x^2-2x+2. concave or convex? For what values of x is f (x)= (x^2−x)e^x. concave or convex? Is f (x)=cosx.
People also ask
How do you analyze concavity if n 1 is separable?
How do you determine the concavity of a function?
How do you determine the concavity of a quadratic function?
How do you know if a function is concave up or down?
When is a twice-differentiable function f concave down?
What is the concavity of a function?
Proof. If fis concave then for any x;x 2C, x6= x, and any 2(0;1), f( x+ (1 )x) f(x) + (1 )f(x), or, dividing by and rearranging, f(x) f(x) f(x + (x x)) f(x) : Taking the limit of the right-hand side as #0 and rearranging yields inequality (7). Conversely, consider any a;b2C, take any 2(0;1), and let x = a+(1 )b.
