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Câu hỏi: Xét hàm số $f\left( x \right)=a\ln \left( x+\sqrt{{{x}^{2}}+1} \right)+b\sin 4x+c{{.10}^{x}}$. Với a, b, c là những hằng số. Biết $f\left( \log \left( \log e \right) \right)+f\left( \log \left( \ln 10 \right) \right)=4$. Giá trị của c nằm trong khoảng nào?
A. $\left( 1;\dfrac{3}{2} \right)$.
B. $\left( 0;1 \right)$.
C. $\left( \dfrac{3}{2};2 \right)$.
D. $\left( 2;3 \right)$.
A. $\left( 1;\dfrac{3}{2} \right)$.
B. $\left( 0;1 \right)$.
C. $\left( \dfrac{3}{2};2 \right)$.
D. $\left( 2;3 \right)$.
Ta có: $f\left( \log \left( \ln 10 \right) \right)=f\left( \log \left( \dfrac{1}{\log e} \right) \right)=f\left[ -\log \left( \log e \right) \right]$
Mặt khác $f\left( -x \right)=a\ln \left( \sqrt{{{x}^{2}}+1}-x \right)+b\sin \left( -4x \right)+c{{.10}^{-x}}=a\ln \dfrac{1}{\sqrt{{{x}^{2}}+2}+x}-b\sin 4x+c{{.10}^{-x}}$
$=-a\ln \left( x+\sqrt{{{x}^{2}}+1} \right)-b\sin 4x+c{{.10}^{-x}}\Rightarrow f\left( x \right)+f\left( -x \right)=c\left( {{10}^{x}}+{{10}^{-x}} \right)$
Đặt ${{x}_{0}}=\log \left( \log e \right)$, khi đó $f\left( \log \left( \log e \right) \right)+f\left( \log \left( \ln 10 \right) \right)=c\left( {{10}^{{{x}_{0}}}}+{{10}^{-{{x}_{0}}}} \right)=4$
$\Leftrightarrow c=\dfrac{4}{{{10}^{{{x}_{0}}}}+{{10}^{-{{x}_{0}}}}}=\dfrac{4}{\log e+\dfrac{1}{\log e}}\approx 1,46$.
Mặt khác $f\left( -x \right)=a\ln \left( \sqrt{{{x}^{2}}+1}-x \right)+b\sin \left( -4x \right)+c{{.10}^{-x}}=a\ln \dfrac{1}{\sqrt{{{x}^{2}}+2}+x}-b\sin 4x+c{{.10}^{-x}}$
$=-a\ln \left( x+\sqrt{{{x}^{2}}+1} \right)-b\sin 4x+c{{.10}^{-x}}\Rightarrow f\left( x \right)+f\left( -x \right)=c\left( {{10}^{x}}+{{10}^{-x}} \right)$
Đặt ${{x}_{0}}=\log \left( \log e \right)$, khi đó $f\left( \log \left( \log e \right) \right)+f\left( \log \left( \ln 10 \right) \right)=c\left( {{10}^{{{x}_{0}}}}+{{10}^{-{{x}_{0}}}} \right)=4$
$\Leftrightarrow c=\dfrac{4}{{{10}^{{{x}_{0}}}}+{{10}^{-{{x}_{0}}}}}=\dfrac{4}{\log e+\dfrac{1}{\log e}}\approx 1,46$.
Đáp án A.