3.
![f'(x)= \frac{4}{7}* \frac{7}{4}x^{ \frac{7}{4}- \frac{4}{4} }+ \frac{8}{5}* \frac{5}{4}x^{ \frac{5}{4}- \frac{4}{4} }-12=x^{ \frac{3}{4} }+2x^{ \frac{1}{4} }-12](https://tex.z-dn.net/?f=f%27%28x%29%3D+%5Cfrac%7B4%7D%7B7%7D%2A+%5Cfrac%7B7%7D%7B4%7Dx%5E%7B+%5Cfrac%7B7%7D%7B4%7D-+%5Cfrac%7B4%7D%7B4%7D++%7D%2B+%5Cfrac%7B8%7D%7B5%7D%2A+%5Cfrac%7B5%7D%7B4%7Dx%5E%7B+%5Cfrac%7B5%7D%7B4%7D-+%5Cfrac%7B4%7D%7B4%7D++%7D-12%3Dx%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D%2B2x%5E%7B+%5Cfrac%7B1%7D%7B4%7D+%7D-12++++)
![x^{ \frac{3}{4} }+2x-12=0 \\ (x^{ \frac{1}{4} })^3+2x^{ \frac{1}{4} }-12=0 \\ t=x^{ \frac{1}{4} } \\ t^3+2t-12=0](https://tex.z-dn.net/?f=x%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D%2B2x-12%3D0+%5C%5C+%0A%28x%5E%7B+%5Cfrac%7B1%7D%7B4%7D+%7D%29%5E3%2B2x%5E%7B+%5Cfrac%7B1%7D%7B4%7D+%7D-12%3D0+%5C%5C+%0At%3Dx%5E%7B+%5Cfrac%7B1%7D%7B4%7D++%7D+%5C%5C+%0At%5E3%2B2t-12%3D0)
При t=2
2³+2*2-12=8+4-12=0
t=2 - это один из корней уравнения.
Делим t³+2t-12 на t-2, получаем
(t³ + 2t-12) : (t-2) = t²+2t+6
(t-2)(t²+2t+6)=0
t-2=0 t²+2t+6=0
t=2 D=2² -4*6=4-24=-20<0
нет действительных корней
![x^{ \frac{1}{4} }=2 \\ (x^{ \frac{1}{4} })^4=2^4 \\ x=16](https://tex.z-dn.net/?f=x%5E%7B+%5Cfrac%7B1%7D%7B4%7D+%7D%3D2+%5C%5C+%0A%28x%5E%7B+%5Cfrac%7B1%7D%7B4%7D+%7D%29%5E4%3D2%5E4+%5C%5C+%0Ax%3D16)
Ответ: 16.
1.
![y=4(3x+1)^{ \frac{3}{4} }-4.5x \\ y'=4* \frac{3}{4} (3x+1)^{ \frac{3}{4}- \frac{4}{4} }*3-4.5=9(3x+1)^{- \frac{1}{4} }-4.5 \\ \\ 9(3x+1)^{- \frac{1}{4} }-4.5=0 \\ (3x+1)^{- \frac{1}{4} }-0.5=0 \\ (3x+1)^{- \frac{1}{4} }= \frac{1}{2} \\ ((3x+1)^{- \frac{1}{4} })^{-4}=( \frac{1}{2} )^{-4} \\ \\ 3x+1=2^4 \\ 3x=16-1 \\ 3x=15 \\ x=5](https://tex.z-dn.net/?f=y%3D4%283x%2B1%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D-4.5x+%5C%5C+%0Ay%27%3D4%2A+%5Cfrac%7B3%7D%7B4%7D+%283x%2B1%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D-+%5Cfrac%7B4%7D%7B4%7D++%7D%2A3-4.5%3D9%283x%2B1%29%5E%7B-+%5Cfrac%7B1%7D%7B4%7D+%7D-4.5+%5C%5C+%0A+%5C%5C+%0A9%283x%2B1%29%5E%7B-+%5Cfrac%7B1%7D%7B4%7D+%7D-4.5%3D0+%5C%5C+%0A%283x%2B1%29%5E%7B-+%5Cfrac%7B1%7D%7B4%7D+%7D-0.5%3D0+%5C%5C+%0A%283x%2B1%29%5E%7B-+%5Cfrac%7B1%7D%7B4%7D+%7D%3D+%5Cfrac%7B1%7D%7B2%7D+%5C%5C+%0A%28%283x%2B1%29%5E%7B-+%5Cfrac%7B1%7D%7B4%7D+%7D%29%5E%7B-4%7D%3D%28+%5Cfrac%7B1%7D%7B2%7D+%29%5E%7B-4%7D+%5C%5C++%5C%5C+%0A3x%2B1%3D2%5E4+%5C%5C+%0A3x%3D16-1+%5C%5C+%0A3x%3D15+%5C%5C+%0Ax%3D5+)
+ -
---------- 5 --------------
x=5 - точка максимума
При x∈(-∞; 5] функция возрастает.
При x∈[5; +∞) функция убывает.
б) При х=0
![y=4(3*0+1)^{ \frac{3}{4} }-4.5*0=4*1^{ \frac{3}{4} }=4](https://tex.z-dn.net/?f=y%3D4%283%2A0%2B1%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D-4.5%2A0%3D4%2A1%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D%3D4)
y=4 - наименьшее значение функции
При х=5
![y=4(3*5+1)^{ \frac{3}{4} }-4.5*5=4*16^{ \frac{3}{4} }-22.5=4*(2^4)^{ \frac{3}{4} }-22.5= \\ =4*8-22.5=32-22.5=9.5](https://tex.z-dn.net/?f=y%3D4%283%2A5%2B1%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D-4.5%2A5%3D4%2A16%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D-22.5%3D4%2A%282%5E4%29%5E%7B+%5Cfrac%7B3%7D%7B4%7D+%7D-22.5%3D+%5C%5C+%0A%3D4%2A8-22.5%3D32-22.5%3D9.5)
у=9,5 - наибольшее значение функции
2.
![y=2.5x^{ \frac{6}{5} }-2x \\ y'=2.5* \frac{6}{5}x^{ \frac{6}{5}- \frac{5}{5} }-2=3x^{ \frac{1}{5} }-2](https://tex.z-dn.net/?f=y%3D2.5x%5E%7B+%5Cfrac%7B6%7D%7B5%7D+%7D-2x+%5C%5C+%0Ay%27%3D2.5%2A+%5Cfrac%7B6%7D%7B5%7Dx%5E%7B+%5Cfrac%7B6%7D%7B5%7D-+%5Cfrac%7B5%7D%7B5%7D++%7D-2%3D3x%5E%7B+%5Cfrac%7B1%7D%7B5%7D+%7D-2+)
y=4x+1
k=4
y'=k
![3x^{ \frac{1}{5} }-2=4 \\ 3x^{ \frac{1}{5} }=4+2 \\ 3x^{ \frac{1}{5} }=6 \\ x^{ \frac{1}{5} }=2 \\ x=2^5=32](https://tex.z-dn.net/?f=3x%5E%7B+%5Cfrac%7B1%7D%7B5%7D+%7D-2%3D4+%5C%5C+%0A3x%5E%7B+%5Cfrac%7B1%7D%7B5%7D+%7D%3D4%2B2+%5C%5C+%0A3x%5E%7B+%5Cfrac%7B1%7D%7B5%7D+%7D%3D6+%5C%5C+%0Ax%5E%7B+%5Cfrac%7B1%7D%7B5%7D+%7D%3D2+%5C%5C+%0Ax%3D2%5E5%3D32)
x=x₀=2⁵=32
![y(x_{0})=2.5*(2^5)^{ \frac{6}{5} }-2*32=2.5*2^6-64=2.5*64-64=96](https://tex.z-dn.net/?f=y%28x_%7B0%7D%29%3D2.5%2A%282%5E5%29%5E%7B+%5Cfrac%7B6%7D%7B5%7D+%7D-2%2A32%3D2.5%2A2%5E6-64%3D2.5%2A64-64%3D96)
y=96+4(x-32)=96+4x-128=4x-32
y=4x-32 - уравнение касательной.
Ответ: у=4х-32.