Q. 7 - Calculus

Question

If a continuous, differentiable function f has zeroes at x = −4, x = 1, and x = 2, what can you say about f ' on [−4, 2]?

Step-by-Step Solution

Answer

The function f is continuous and differentiable on $\left[-4,2\right]$ but not  satisfied all the conditions .

Step 1. Given information .

Consider the given point $\left[-2,4\right]$ .

Step 2. Find the function .

The function $f\left(x\right)= x^3+x^2-10x+8$ has roots $x=-4 , x=1 . x=2$.

The given function f(x) is continuous and differentiable at given points .

Step 3. Find the derivative .

The derivative of the given function is .

$$\begin{gathered} f\left(x\right)=x^3+x^2-10x+8 \\ f\text{'}\left(x\right)=3x^2+2x-10 \end{gathered}$$

Putting the values $x=-4 x=2$

$$\begin{gathered} f\text{'}\left(2\right)=3{\left(2\right)}^2+2\left(2\right)-10 \\ f\text{'}\left(2\right)=6 \end{gathered}$$

$$\begin{gathered} f\text{'}\left(-4\right)=3{\left(-4\right)}^2+2\left(-4\right)-10 \\ f\text{'}\left(-4\right)=30 \end{gathered}$$

The given function $f\text{'}\left(x\right)$ not equal to zero so the given points are the critical points of the function but not local extremum .