p. 467

89.

a)

Plot[
       { Sin[x],  Log[2 + Sin[x]], Log[4 + Sin[x]], Log[8 + Sin[x]], 
        Log[20 + Sin[x]], Log[50 + Sin[x]] },  {x, 0, 23}, 
      PlotStyle -> {Hue[ .1], Hue[ .2], Hue[ .3], Hue[ .4], Hue[ .5], 
          Hue[ .6]}]

b)

D[ Log[a + Sin[x]],  x]

The largest the numerator can get (in absolute value) is 1, and the smallest the denominator can get is a  -  1, so the above derivative is always less than or equal to (1/(a - 1)). For large vales of a, this number is small, for example, (1/(50 - 1)) = 0.0204081632653061229`. When the derivative changes only a small amount like this, the function itself cannot change much and the graph looks "flat".

90.

f[x_]  :=  (Sqrt[x]) -  Log[x]

f''[x]

Solve[f''[x]  ==  0]

Plot[f''[x], {x, 15, 17}]

Since the sign of f''[x]changes from positive to negative at x  =  16, the function has an inflection point there. This is a little hard to see just from the plot:

Plot[f[x], {x,  .1, 50}]