- Define the following functions:
for . Prove that for and the inequalities
hold. Proceed by examining the signs of the derivatives and . When these are equalities.
- Draw the graphs of the functions
Interpret the inequalities in part (a) geometrically.
- First, we compute the derivatives of the functions,
Then, considering the derivative of ,
Therefore, the function has a minimum at . Since we can directly evaluate , this means for and . Therefore, .
Next, looking at the derivative of ,
Therefore, the function has a minimum at . Since this implies for and . Thus,
Putting these two pieces together we have established the requested inequalities:
The graph of the two functions is:
The inequalities in part (a) imply that the graph of must lie strictly between the graphs of and shown above (and so ).