Explicație pas cu pas:
[tex]f(x) = x(5+2x²)+2(1 – x)³ = 5x + 2{x}^{3} + 2 - 6x + 6{x}^{2} - 2{x}^{3} = 6 {x}^{2} - x + 2[/tex]
[tex]f'(x)=(6{x}^{2} - x + 2)'=12x-1[/tex]
[tex]f'(x)=0 = >12x-1 = 0 = > x = \frac{1}{12} [/tex]
[tex]x < \frac{1}{12} = > f'(x) < 0[/tex]
[tex]x > \frac{1}{12} = > f'(x) > 0[/tex]
[tex]f(\frac{1}{12})=6 \times {( \frac{1}{12})}^{2} - \frac{1}{12} + 2 = \frac{47}{24} [/tex]
=> funcția are un minim
[tex](\frac{1}{12} ; \frac{47}{24})[/tex]
punct de extrem:
[tex]x = \frac{1}{12} [/tex]
