1. ## Sketching Functions

A function has a local maximum at x=-2, a local maximum at x=6 and a local minimum at x=1.

1a)What does this information tell you about the function?
1b) What is a possible function that fits these characteristics?

Can anyone help me with this please?

2. Originally Posted by notoriousmc
A function has a local maximum at x=-2, a local maximum at x=6 and a local minimum at x=1.

1a)What does this information tell you about the function?
What do you know about the derivative? What about the possible intervals on which the function is increasing and decreasing?

1b) What is a possible function that fits these characteristics?
There are infinitely many such functions. For simplicity, we'll choose a simple polynomial.

Our polynomial $p(x)$ must have a continuous derivative, so $p'(-2) = p'(1) = p'(6) = 0\text{.}$ Thus, $p'(x) = a(x + 2)(x - 1)(x - 6)$ for some constant $a\neq0$. We have to choose a negative value for $a$ in order for the derivative to be positive and negative in the appropriate intervals, so we'll take $p'(x) = -(x+2)(x-1)(x-6)$. Now, integrate and make an arbitrary choice for the integration constant to get a suitable $p(x)$.