Question

The maximum value of $f\left(x\right)=2x^3-21x^2+36x+20$ in $0\le x\le 2$ $is$

• A. $37$
• B. $44$
• C. $32$
• D. $30$

Answer 1

The correct option is A $37$

$f\left(x\right)=2x^3-21x^2+36x+20$
$f^{\prime }\left(x\right)=6(x^2-7x+6)$
For maxima or minima
$f^{\prime }\left(x\right)=0$
$\Rightarrow x=1,6$
$\Rightarrow x=1$ (since 6 does not belongs in the given interval)
Now, $f^{\prime \prime }\left(x\right)=12x-42$
$\Rightarrow f^{\prime \prime }\left(1\right)=-30<0$
f(x) has a maximum at $x=1$.
$f\left(1\right)=37$
Also, $f\left(0\right)=20$ and $f\left(2\right)=24$
Hence,f(x) has a maximum at $x=1$.