open interval and tangent line

[lawilson8510]

Suppose that I is an open interval and that f "(x) is greater than or equal to 0 for all x in I. If b is in I, then show that the part of the graph of f on I is never below the tangent line to the graph at (b, f(b)).

 

[pka]

Suppose that I is an open interval and that f "(x) is greater than or equal to 0 for all x in I. If b is in I, then show that the part of the graph of f on I is never below the tangent line to the graph at (b, f(b)).

Let \(\displaystyle \varphi (x) = f(x) - f'(b)\left( {x - b} \right) - f(b)\), now you want to show that \(\displaystyle \varphi (x) \ge 0\).
Here is what we must know:
\(\displaystyle \varphi (b)=?\)
\(\displaystyle \varphi '(x) = f'(x) - f'(b)\)
\(\displaystyle \varphi ''(x) = f''(x) \geqslant 0\)

Now that is enough to prove it if you can put it together.