# 2010-18 Limit of a differentiable function

Let f be a differentiable function. Prove that if $$\lim_{x\to\infty} (f(x)+f'(x))=1$$, then $$\lim_{x\to\infty} f(x)=1$$.

GD Star Rating

# 2009-10 x and cos x

Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.

GD Star Rating
Find all real numbers $$\lambda$$ and the corresponding functions $$f$$ such that the equation
$$\displaystyle \int_0^1 \min(x,y) f(y) \,dy=\lambda f(x)$$
has a non-zero solution $$f$$ that is continuous on the interval [0,1].