# 2012-14 Equation with Integration

Determine all continuous functions $$f:(0,\infty)\to(0,\infty)$$ such that $\int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx$ for all $$t>0$$.

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# 2011-18 Continuous Function and Differentiable Function

Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if

|g(xf(x)+c g′(x)|≤|g(x)| for all x∈I,

then g(x)=0 for all x∈I.

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Let f be a continuous function on [0,1]. Prove that $\lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).$