Questions tagged [mean-value-theorem]

Prove if $f(x)$ has 3 distinct solutions on $[a,b]$ and $f(x)$ has a continuous second derivative on $[a,b]$, then $f''(x)+2f(x)$ has solutions on $[a,b]$. I'm stuck for ideas on this problem. Where can I start? Is it possible to generalize with…

Let $A = [0,1]^2$ and $f(x,y) = (e^x cos y, e^x sin y)$. Using the Mean Value Theorem, determine the maximum value of $||f(x,y)||$ where $(x,y) \in A$. Explain why the maximum is attained on A. I have concluded that the maximum is attained on A…

let $f:[0,10]\to[10,20]$ be a continuous and twice differentiable function such that $f(0)=10$ and $f(10)=20$. Suppose $|f'(x)| \leq 1$ for all $x \in[0,10]$. Then, the value of $f"(5)$ is? This is my crude way of doing it I thought of it like…

I'm studying Stoker's "Differential geometry" and Hormander's "The analysis of linear partial differential operators I, distribution theory and fourier analysis" on my own. On p.10, Stoker says that, for a vector function $\vec x(t)$, we can apply…