Question

The function $f\left(x\right)=x^3-6x^2+9x+25$ has

• A. a maxima at $x=1$ and a minima at $x=3$
• B. a maxima at $x=3$ and a minima at $x=1$
• C. no maxima, but a minima at $x=3$
• D. a maxima at $x=1$, but no minima

Answer 1

The correct option is A a maxima at $x=1$ and a minima at $x=3$

$f\left(x\right)=x^3-6x^2+9x+25$
$f^{\prime }\left(x\right)=3x^2-12x+9$
For critical points:
$f^{\prime }\left(x\right)=0$
$x^2-4x+3=0$
$\left(\right(x-3\left)\right(x-1)=0$
$x=3,1$
$f^{\prime \prime }\left(x\right)=6x-12$
$f^{\prime \prime }\left(x\right)$ at $x=3$ is: $f^{\prime \prime }\left(3\right)=6\times 3-12$
$f^{\prime \prime }\left(3\right)=6>0$
$\because f^{\prime \prime }\left(x\right)$ at $x=3>0$, the function has a minima at $x=3$
$f^{\prime \prime }\left(x\right)$ at $x=1$ is: $f^{\prime \prime }\left(1\right)=6\times 1-12$
$f^{\prime \prime }\left(1\right)=-6<0$
$\because f^{\prime \prime }\left(x\right)$ at $x=1$ is $<0$, the function has a maxima at $x=1$.