Question

Find the maximum and minimum value of function $f\left(x\right)=x^3-6x^2+9x+1$ in the interval $x=0\text{to}x=5$.

• A. 21 and 1
• B. 20 and 2
• C. 16 and 14
• D. 18 and 2

Answer 1

The correct option is A 21 and 1

First, isolate the possible values of $x$ at which maximum and minimum value of function possible which is $x=0$ and $x=5.$

Now find out the critical points by using
$f^{\prime }\left(x\right)=0$
$f^{\prime }\left(x\right)=3x^2-12x+9=0$
$3(x-1)(x-3)=0$
$x=1,3$

So to find out the maximum and minimum value, just calculate the value of function at $x=0,1,3,5$ and compare.
$\begin{array}{ccccc}x& 0& 1& 3& 5\\ f\left(x\right)& 1& 5& 1& 21\end{array}$

Hence, maximum value is $21$ and minimum value is $1$.