GRE Math Subject

We provide tutoring on GRE Math Subject Test and other specific exams or topics (including business mathematics) where assistance is needed as the student prepares for graduate school.

Example:

Which of the following statements is FALSE?
A. For a continuous function $f$ on $(a,b)$. Let $y\in (a,b)$ be a constant. If, for $t\in (a,b)$, $g(t)={\int }_y^{t}f(x)dx$, then $\underset{h\to 0}{lim}\frac{g(t+h)-g(t)}{h}=f(t)$
B. For a continuous function $f$ on $[a,b]$, there exists $x\in [a,b]$ such that $f(x)\ge f(t)\text{for all}t\in [a,b]$
C. The function $f(x)=\frac{1}{x}$ is continuous on $(0,\mathrm{\infty })$ but not uniformly continuous on $(0,\mathrm{\infty })$.
D. ${\int }_0^2(\lfloor x\rfloor -\lceil x\rceil )e^{-x}dx=e^{-2}-1$
E. Let $f$ be continuous on $(a,b)$ and suppose that $x,y\in (a,b)$ with $x\ne y$. Then, there exists $t\in (a,b)$ such that $f(t)=\frac{f(x)-f(y)}{x-y}$.

Solution: E is False: Try $f(x)=\frac{1}{x}\text{on}(0,1)$, and use $x=\frac{1}{2},y=\frac{1}{4}$. There is no   $t\in (0,1)$ for which $f(t)=\frac{f(x)-f(y)}{x-y}$
because $\frac{f(x)-f(y)}{x-y}$ is negative ($=-8$).