## TPTP Problem File: RAL056^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL056^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Quadratic functions)
% Problem  : Kyushu University, 2003, Humanities Course, Problem 3
% Version  : [Mat16] axioms : Especial.
% English  : Let a, b, and c be real numbers, and assume that a > 0.  Define
%            f(x)= a x^2 + b x + c. For the real number p, let g(p) be the
%            maximum value of the function p x - f(x). (1) When the functions
%            y = f(x) and y = g(x) match each other, find the value of f(x).
%            (2) For the real number x, let h(x) be the maximum value of the
%            function of p: x p - g(p). Find the value of h(x). (3) Prove that
%            the necessary and sufficient condition for the straight line
%            y = p x + q to be in contact with the graph of y = f(x) at the
%            point (t, f(t)) is g(p)= p t - f(t) and q = - g(p).

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Kyushu-2003-Bun-3.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3489 (   0 unit;1203 type;   0 defn)
%            Number of atoms       : 45345 (2212 equality;22717 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39635 ( 104   ~; 233   |;1175   &;35996   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1250 (1203   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8064 (  66 sgn;7087   !; 434   ?; 407   ^)
%                                         (8064   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1981 (   6 prd;   9 fun;  23 num;1943 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Yiyang Zhan; Generated: 2014-05-23
%            ^ [V_abc_dot_0: ( ''ListOf'' @ \$real )] :
%              ( V_abc_dot_0
%              = ( 'cons/2' @ \$real @ ( \$quotient @ 1.0 @ 2.0 ) @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ 0.0 @ ( 'nil/0' @ \$real ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,(
'a/0': \$real )).

thf('b/0_type',type,(
'b/0': \$real )).

thf('c/0_type',type,(
'c/0': \$real )).

thf('x/0_type',type,(
'x/0': \$real )).

thf(p1_m_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ \$real )
@ ( ^ [V_abc: ( 'ListOf' @ \$real )] :
? [V_a: \$real,V_b: \$real,V_c: \$real,V_f: 'R2R'] :
( ( \$less @ 0.0 @ V_a )
& ( V_f
= ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ V_c @ ( 'cons/2' @ \$real @ V_b @ ( 'cons/2' @ \$real @ V_a @ ( 'nil/0' @ \$real ) ) ) ) ) )
& ! [V_p: \$real,V_gp: \$real] :
( ( 'maximum/2'
@ ( 'set-by-def/1' @ \$real
@ ( ^ [V_v: \$real] :
? [V_x: \$real] :
( V_v
= ( \$difference @ ( \$product @ V_p @ V_x ) @ ( 'funapp/2' @ V_f @ V_x ) ) ) ) ) @ V_gp )
=> ( ( 'funapp/2' @ V_f @ V_p )
= V_gp ) )
& ( V_abc
= ( 'cons/2' @ \$real @ V_a @ ( 'cons/2' @ \$real @ V_b @ ( 'cons/2' @ \$real @ V_c @ ( 'nil/0' @ \$real ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------
```