Show that $f(x)=x^2$ is not a decreasing function between $a=-10$ and $b=2$ by evaluating $f\,'(1)$.
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Problem:
Show that $f(x)=x^2$ is not a decreasing function between $a=-10$ and $b=2$ by evaluating $f\,'(1)$.
Answer:
It is true that $f(x)=x^2$ is not a decreasing function between $a=-10$ and $b=2$.
Solution:
Since $f(-10)=100$ and $f(2)=4$ as we go from $-10$ to $2$ (left to right) the function decreases. Here is a picture of the function:
However, we know that $f\,'(x)=2x$ and that $f\,'(1)=2$ which means that $f(x)$ has a positive slope at $x=1$. Therefore,the function is both decreasing and increasing and we can't say that it is a decreasing function between $-10$ and $2$.
