Question

graph the following function. estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima. f(x)=3x^2

Explanation:

Step1: Recall derivative rules

The derivative of $f(x)=3x^{2}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f^\prime(x)=6x$.

Step2: Find critical points

Set $f^\prime(x) = 0$. So, $6x=0$, which gives $x = 0$ as the critical point.

Step3: Test intervals

Choose a test point in the interval $(-\infty,0)$, say $x=-1$. Then $f^\prime(-1)=6\times(-1)=-6<0$, so $f(x)$ is decreasing on $(-\infty,0)$.
Choose a test point in the interval $(0,\infty)$, say $x = 1$. Then $f^\prime(1)=6\times1=6>0$, so $f(x)$ is increasing on $(0,\infty)$.

Step4: Determine relative extrema

Since the function changes from decreasing to increasing at $x = 0$, $f(0)=3\times0^{2}=0$ is a relative minimum.

To graph $y = 3x^{2}$, it is a parabola opening upwards with vertex at the origin $(0,0)$.

Answer:

The function $f(x)=3x^{2}$ is decreasing on the interval $(-\infty,0)$ and increasing on the interval $(0,\infty)$. The relative minimum is $0$ at $x = 0$.