graph the following piecewise equation and define the intervals for which the function is increasing…

Explanation:

Step1: Analyze $y = x^{3}$ for $x\leq1$

The derivative of $y=x^{3}$ is $y' = 3x^{2}$. Since $y'=3x^{2}\geq0$ for all real - x, and $y' = 0$ only at $x = 0$, the function $y=x^{3}$ is increasing on the interval $(-\infty,1]$.

Step2: Analyze $y=\frac{1}{4}x^{2}$ for $1\lt x\lt4$

The derivative of $y=\frac{1}{4}x^{2}$ is $y'=\frac{1}{2}x$. When $y'=\frac{1}{2}x>0$ for $x>0$, so for the interval $(1,4)$, the function $y = \frac{1}{4}x^{2}$ is increasing.

Step3: Analyze $y = 4$ for $x\geq5$

The function $y = 4$ is a constant function. Its derivative $y'=0$. So it is constant on the interval $[5,\infty)$.

Step4: Determine non - defined intervals

The function is defined for all real numbers. There are no non - defined intervals.

Answer:

• Graph:
  • For $y = x^{3}$ when $x\leq1$, we can plot points such as $(-1,-1),(0,0),(1,1)$.
  • For $y=\frac{1}{4}x^{2}$ when $1\lt x\lt4$, we can plot points like $(2,1),(3,\frac{9}{4})$.
  • For $y = 4$ when $x\geq5$, we have a horizontal line at $y = 4$ starting from $x = 5$.
• Intervals of the piece - wise function:
  • $y=x^{3}$ for $x\in(-\infty,1]$; $y=\frac{1}{4}x^{2}$ for $x\in(1,4)$; $y = 4$ for $x\in[5,\infty)$.
• Intervals of increase: $(-\infty,1]\cup(1,4)=(-\infty,4)$.
• Intervals of decrease: None.
• Intervals where the function remains constant: $[5,\infty)$.
• Intervals for which the function is not defined: None.