JEE Advance - Mathematics (2017 - Paper 2 Offline - No. 8)
f : R $$ \to $$ R is a differentiable function such that f'(x) > 2f(x) for all x$$ \in $$R, and f(0) = 1 then
f(x) > e2x in (0, $$\infty $$)
f'(x) < e2x in (0, $$\infty $$)
f(x) is increasing in (0, $$\infty $$)
f(x) is decreasing in (0, $$\infty $$)
Explicació
$$f'(x) > 2f(x)$$
$$ \Rightarrow {{dy} \over y} > 2dx$$
$$ \Rightarrow \int_1^{f(x)} {{{dy} \over y} > 2\int_0^x {dx} } $$
$$\ln (f(x)) > 2x$$
$$ \therefore $$ $$f(x) > {e^{2x}}$$
Also, as $$f'(x) > 2f(x)$$
$$ \therefore $$ $$f'(x) > 2{c^{2x}} > 0$$
$$ \Rightarrow {{dy} \over y} > 2dx$$
$$ \Rightarrow \int_1^{f(x)} {{{dy} \over y} > 2\int_0^x {dx} } $$
$$\ln (f(x)) > 2x$$
$$ \therefore $$ $$f(x) > {e^{2x}}$$
Also, as $$f'(x) > 2f(x)$$
$$ \therefore $$ $$f'(x) > 2{c^{2x}} > 0$$