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C(x)=0.04x^(2)

Over what interval is each function increasing and over what interval is each function decreasing?
SEE EXAMPLE 3
20.





x

f(x)=-0.3x^(2)

(x,y)


-2
-1.2

(-2,-1.2)


-1
-0.3

(-1,-0.3)


0
0

(0,0)


1
-0.3

(1,-0.3)


2
-1.2

(2,-1.2)

1818. C(x)=0.04x2 C(x)=0.04 x^{2} \newlineOver what interval is each function increasing and over what interval is each function decreasing?\newlineSEE EXAMPLE 33\newline2020.\newline\begin{tabular}{|c|c|c|}\newline\hlinex x & f(x)=0.3x2 f(x)=-0.3 x^{2} & (x,y) (x, y) \\\newline\hline2-2 & 1-1.22 & (2,1.2) (-2,-1.2) \\\newline\hline1-1 & 0-0.33 & (1,0.3) (-1,-0.3) \\\newline\hline 00 & 00 & (0,0) (0,0) \\\newline\hline 11 & 0-0.33 & (1,0.3) (1,-0.3) \\\newline\hline 22 & 1-1.22 & (2,1.2) (2,-1.2) \\\newline\hline\newline\end{tabular}

Full solution

Q. 1818. C(x)=0.04x2 C(x)=0.04 x^{2} \newlineOver what interval is each function increasing and over what interval is each function decreasing?\newlineSEE EXAMPLE 33\newline2020.\newline\begin{tabular}{|c|c|c|}\newline\hlinex x & f(x)=0.3x2 f(x)=-0.3 x^{2} & (x,y) (x, y) \\\newline\hline2-2 & 1-1.22 & (2,1.2) (-2,-1.2) \\\newline\hline1-1 & 0-0.33 & (1,0.3) (-1,-0.3) \\\newline\hline 00 & 00 & (0,0) (0,0) \\\newline\hline 11 & 0-0.33 & (1,0.3) (1,-0.3) \\\newline\hline 22 & 1-1.22 & (2,1.2) (2,-1.2) \\\newline\hline\newline\end{tabular}
  1. Find Derivative: To determine where the function C(x)=0.04x2C(x)=0.04x^{2} is increasing or decreasing, we need to find its first derivative, C(x)C'(x), which will give us the rate of change of the function.
  2. Calculate Derivative: Calculate the first derivative of C(x)=0.04x2C(x)=0.04x^{2} with respect to xx.\newlineC(x)=ddx(0.04x2)=0.08xC'(x) = \frac{d}{dx} (0.04x^{2}) = 0.08x
  3. Analyze Derivative: Analyze the first derivative to determine where it is positive (function is increasing) and where it is negative (function is decreasing).\newlineSince C(x)=0.08xC'(x) = 0.08x, the derivative is positive when x>0x > 0 and negative when x<0x < 0.
  4. Conclude Intervals: Conclude the intervals of increase and decrease for the function C(x)C(x). The function C(x)C(x) is increasing on the interval (0,)(0, \infty) and decreasing on the interval (,0)(-\infty, 0).

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