Suppose along the line xk +adk ,a> 0, the function f (xk +adk) is unimodal and differentiable....

Suppose along the line xk +αdk
,α> 0, the function f (xk +αdk) is unimodal and differentiable.
Let αk be the minimizing value of α. Show that if any αk >
αk is selected to define xk+1 = xk +αkdk, then pT k qk > 0. (Refer
to Sect. 10.3.)

Let {Hk}, k = 0, 1, 2 ... be the sequence
of matrices generated by the Davidon–Fletcher–Powell method applied, without
restarting, to a function f having continuous second partial derivatives.
Assuming that there is a > 0,A> 0 such that for all k we have Hk −
aI and AI − Hk positive definite and the corresponding sequence of xk’s
is bounded, show that the method is globally convergent.