7.
![f(x)=( \sqrt{x} + \frac{1}{ \sqrt{x} })^{40}=(x^{ \frac{1}{2} } +x^{ -\frac{1}{2} })^{40}](https://tex.z-dn.net/?f=f%28x%29%3D%28%20%5Csqrt%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7B%20%5Csqrt%7Bx%7D%20%7D%29%5E%7B40%7D%3D%28x%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%2Bx%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D%29%5E%7B40%7D)
![f'(x)=((x^{ \frac{1}{2} } +x^{ -\frac{1}{2} })^{40})'=40(x^{ \frac{1}{2} } +x^{ -\frac{1}{2} })^{39}*( \frac{1}{2}x^{ -\frac{1}{2} }- \frac{1}{2}x^{ -\frac{3}{2} } )=](https://tex.z-dn.net/?f=f%27%28x%29%3D%28%28x%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%2Bx%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D%29%5E%7B40%7D%29%27%3D40%28x%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%2Bx%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D%29%5E%7B39%7D%2A%28%20%5Cfrac%7B1%7D%7B2%7Dx%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D-%20%5Cfrac%7B1%7D%7B2%7Dx%5E%7B%20-%5Cfrac%7B3%7D%7B2%7D%20%7D%20%29%3D)
![=20(x^{ \frac{1}{2} } +x^{ -\frac{1}{2} })^{39}*( x^{ -\frac{1}{2} }- x^{ -\frac{3}{2} } )](https://tex.z-dn.net/?f=%3D20%28x%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%2Bx%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D%29%5E%7B39%7D%2A%28%20x%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D-%20x%5E%7B%20-%5Cfrac%7B3%7D%7B2%7D%20%7D%20%29)
![f'(1)=20(1^{ \frac{1}{2} } +1^{ -\frac{1}{2} })^{39}*( 1^{ -\frac{1}{2} }- 1^{ -\frac{3}{2} } )=0](https://tex.z-dn.net/?f=f%27%281%29%3D20%281%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%2B1%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D%29%5E%7B39%7D%2A%28%201%5E%7B%20-%5Cfrac%7B1%7D%7B2%7D%20%7D-%201%5E%7B%20-%5Cfrac%7B3%7D%7B2%7D%20%7D%20%29%3D0)
8. f(x)=(x-2)²(x+4)=(x²-4x+4)(x+4)=x³+4x²-4x²-16x+4x+16=x³-12x+16
f'(x)=3x²-12≤0
3x²≤12
x²≤4
x∈[-2;2]
3. (5x+4)-(3x-6)=4.
5x+4-(3x-6)=4
5x-3x+6=0
2x+6=0
2x=-6
x=-3
Если Боря отдаст Вове 3 книги у него останется 3 книги. Потом Вова отдаст Боре тоже 3 книги. Тогда у Бори будет 6 книг, а у Вовы 4.