"f(x)=3^x(\\log_2(x)-3)^5e^{x^2}-3x" Domain
"D(f): (0, \\infin)"
If "0<x<8," then "\\log_2(x)-3<0." Hence "f(x)<0" for "0<x<8."
Find the first derivative
"f'(x)=3^x\\cdot\\ln3\\cdot(\\log_2(x)-3)^5e^{x^2}+""+5\\cdot3^x\\cdot{1 \\over x\\ln2}(\\log_2(x)-3)^4e^{x^2}+""+2x\\cdot3^x(\\log_2(x)-3)^5e^{x^2}-3"
Find the second derivative
"f''(x)=3^x\\cdot(\\ln3)^2\\cdot(\\log_2(x)-3)^5e^{x^2}+""+5\\cdot3^x\\cdot{\\ln3 \\over x\\ln2}\\cdot(\\log_2(x)-3)^4e^{x^2}+""+2x\\ln3\\cdot3^x(\\log_2(x)-3)^5e^{x^2}+""+5\\cdot3^x\\cdot{\\ln3 \\over x\\ln2}(\\log_2(x)-3)^4e^{x^2}-""-5\\cdot3^x\\cdot{1 \\over x^2\\ln2}(\\log_2(x)-3)^4e^{x^2}+""+20\\cdot3^x\\cdot{\\ln3 \\over x^2(\\ln2)^2}\\cdot(\\log_2(x)-3)^3e^{x^2}+""+10\\cdot3^x\\cdot{1 \\over \\ln2}(\\log_2(x)-3)^4e^{x^2}+""+2\\cdot3^x(\\log_2(x)-3)^5e^{x^2}""+2x\\cdot\\ln3\\cdot3^x(\\log_2(x)-3)^5e^{x^2}""+10\\cdot3^x\\cdot{1 \\over \\ln2}(\\log_2(x)-3)^4e^{x^2}+""+4x^2\\cdot3^x(\\log_2(x)-3)^5e^{x^2}" If "x>8," then "f''(x)>0." Hence "f'(x)" increases for "x>8."
"f(8)=-24<0,"
"f'(8)=-3<0,"
"f''(8)=0."
"f(8.0000000936436)\\approx-24.0000"
"f'(8.0000000936436)=0.000011>0"
"f(8.00000498410413474204)\\approx-0.0000000729"
"f(8.00000498410413474205)\\approx0.00000004565"
The only root
"x\\approx8.00000498410413474205"
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