let f(x)=-x^3 + 4x^2+2. find an open interval (a,b) on which f increases. as your answer, please input a + b…

let f(x)=-x^3 + 4x^2+2. find an open interval (a,b) on which f increases. as your answer, please input a + b in decimal form with three significant digits after the decimal point.

let f(x)=-x^3 + 4x^2+2. find an open interval (a,b) on which f increases. as your answer, please input a + b in decimal form with three significant digits after the decimal point.

Answer

Explanation:

Step1: Find the derivative of $f(x)$

$f'(x)=-3x^{2}+8x$

Step2: Set $f'(x)>0$ to find where $f(x)$ increases

$-3x^{2}+8x>0$ $x(-3x + 8)>0$ The roots of the equation $-3x^{2}+8x = 0$ are $x = 0$ and $x=\frac{8}{3}$. The solution of the inequality $x(-3x + 8)>0$ is $0<x<\frac{8}{3}$. So the open - interval is $(0,\frac{8}{3})$.

Step3: Calculate $a + b$

$a = 0$, $b=\frac{8}{3}\approx2.667$ $a + b=0+\frac{8}{3}\approx2.667$

Answer:

$2.667$