Question

Find the absolute maximum and minimum values, if they exist, of each function over the indicated interval. A f(x)= e 2x + 3x in the domain (-2,1] 128 in the domain (0,00) = f(x) = x² + - X

Answer 1

-20 22 1 f(1) = 6 +33 E2,1 To find critical points equate f(x) =0 d 229+33) =o +30 e- 2x 73 2 Applying lograthon In (e.2x) en (2/2) -27 = en (72) = 0.405 x = -0.20 € (-2, I second derivative test f"(a) = d (-26 dae +3) f" (7) = -21-22-22) = 4é?X 6-28 at t = -0.20 f"(-0.20) = 1é2(-0.20) =40040 =4X1526 f(-0.20)=6 =670 ::$(8) hois absolute minimum at [ 2= 2=-0.20

- 2X0.20 Amin f (-0.20) +3 (-0.20) 0.40 - e & 0.60 fmin =15-016 2009 00,-) a 2) 418) = 2* + 128 taba 27 128 EO 22 223. 3 2% -12870 x3 128 2 64 x = (64)'s = 4 second dorivative test f" (a= 2 + 2x128 X² f''(4) = 2+2x128 2 2+2 x 128 = 6 = (4) 3 f'(W) =6 70 :f(x) has minimum at x=4 tmin = flu) = 100* +12= 16+32 Amin=48