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scoray [572]
3 years ago
15

10. Determine the open interval(s) on which the function f(x) = 12x - x^3 is increasing.

Mathematics
1 answer:
Gre4nikov [31]3 years ago
3 0

Answer:

\large\boxed{x\in(-2,\ 2)}

Step-by-step explanation:

f(x)=12x-x^3

Calculate the derivative of the function:

f'(x)=\left(12x-x^3\right)'=(12x)'-\left(x^3\right)'=12-3x^2

Calculate the zeros of the derivative:

f'(x)=0\Rightarrow12-3x^2=0\qquad\text{add}\ 3x^2\ \text{to both sides}\\\\3x^2=12\qquad\text{divide both sides by 3}\\\\x^2=4\to x=\pm\sqrt4\\\\x=\pm2

Let's check the derivative sign:

f'(x)>0\iff12-3x^2>0\qquad\text{use the distributive property}\\\\-3(x^2-4)>0\\\\-3(x^2-2^2)>0\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\\\-3(x-2)(x+2)>0\\^{\qquad x=2\qquad x=-2

Sketch a parabola open down

(before the brackets there is the number -3 < 0)

Look at the picture.

Where the derivative is negative, the function decreases.

Where the derivative is positive, the function increases.

Therefore your answer is:

x\in(-2,\ 2)

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