3.       Given f(x, y) = ln(`+`(`*`(2, `*`(x)), `*`(2, `*`(y)))); determine all values that satisfy the MVT on the interval from point  P(`+`(`*`(`/`(1, 2), `*`(exp(`/`(1, 2)))), `-`(`/`(1, 2))), `/`(1, 2));  to point 

restart;  

 

 

 

p := [`+`(`*`(`/`(1, 2), `*`(exp(`/`(1, 2)))), `-`(`/`(1, 2))), `/`(1, 2)];  

p := [`+`(`*`(`/`(1, 2), `*`(exp(`/`(1, 2)))), `-`(`/`(1, 2))), `/`(1, 2)]; (8.3.1)
 

q := [`+`(`*`(`/`(1, 2), `*`(exp(3))), `-`(3)), 3];  

q := [`+`(`*`(`/`(1, 2), `*`(exp(3))), `-`(3)), 3]; (8.3.2)
 

 

`/`(`*`(`+`(p[2], `-`(q[2]))), `*`(`+`(p[1], `-`(q[1]))));  

`+`(`-`(`*`(5, `*`(`/`(`+`(exp(`/`(1, 2)), 5, `-`(exp(3)))))))); (8.3.3)
 

 

.37210; (8.3.4)
 

ln(`+`(`*`(2, `*`(x)), `*`(2, `*`(y)))) = y; `/`(1, `*`(`+`(`-`(1), y, x))); fp(x, y);  

fsolve({y = ln(`+`(`*`(2, `*`(x)), `*`(2, `*`(y)))), `/`(1, `*`(`+`(`-`(1), y, x))) = (`+`(`-`(`*`(5, `*`(`/`(`+`(exp(`/`(1, 2)), 5, `-`(exp(3))))))));  

{x = 1.689304346, y = 1.998058784}; (8.3.5)
 

MVTpt := [1.689304346, 1.998058784];  

MVTpt := [1.689304346, 1.998058784]; (8.3.6)
 

 

 

with(plots); -1 

k := pointplot([p, q, MVTpt]);  

Plot_2d
 

tanline := proc (x) options operator, arrow, function_assign; `+`(1.998082237, `*`(fp(1.689367373, 1.998082237), `*`(`+`(x, `-`(1.689367373))))) end proc;  

tanline := proc (x) options operator, arrow; `+`(1.998082237, `*`(fp(1.689367373, 1.998082237), `*`(`+`(x, `-`(1.689367373))))) end proc; (8.3.7)
 

 

secline := proc (x) options operator, arrow; `+`(3, `-`(`/`(`*`(`+`(`*`(5, `*`(x)), `-`(`*`(`/`(5, 2), `*`(exp(3)))), 15)), `*`(`+`(exp(`/`(1, 2)), 5, `-`(exp(3))))))) end proc; (8.3.8)
 

crvs := implicitplot([y = ln(`+`(`*`(2, `*`(x)), `*`(2, `*`(y)))), y = tanline(x), y = secline(x)], x = -2 .. 10, y = -2 .. 10, color = [black, red, blue]);  

Plot_2d
 

display(k, crvs);  

Plot_2d