Question

Решите неравенство f(x)=12x^3+2x^4

Answer 1

12x^3+2x^4=2x^3(6+х)=0
6+х=0
х=-6
2x^3=0
х=0
ответ х=0 х=-6

Answer 2

To solve the inequality $f(x) > 0$, where $f(x) = 12x^3 + 2x^4$, we need to find the values of $x$ for which $f(x)$ is positive.

Step 1: Factor out common terms. $f(x) = x^3(12 + 2x)$

Step 2: Find the critical points by setting each factor to zero and solving for $x$. $x^3 = 0 \Rightarrow x = 0$ $12 + 2x = 0 \Rightarrow 2x = -12 \Rightarrow x = -6$

Step 3: Use test points to determine the sign of $f(x)$ in different intervals:

For $x < -6$: Choose $x = -7$ $f(-7) = (-7)^3(12 + 2(-7)) = -7^3(12 - 14) = -7^3(-2) = -343 \times (-2) = 686 > 0$

For $-6 < x < 0$: Choose $x = -1$ $f(-1) = (-1)^3(12 + 2(-1)) = -1^3(12 - 2) = -1^3 \times 10 = -10 < 0$

For $x > 0$: Choose $x = 1$ $f(1) = 1^3(12 + 2(1)) = 1^3(12 + 2) = 1^3 \times 14 = 14 > 0$

Step 4: Analyze the signs in each interval.

$f(x) > 0$ when $x < -6$ or $x > 0$.

So, the solution to the inequality $f(x) > 0$ is $x < -6$ or $x > 0$.