Question

The function $f\left(x\right)=4x^5-25x^4+40x^3-10$ has

• A. One maximum and one minimum
• B. One maximum and two minimums
• C. Two maximums, one minimum
• D. Two maximums, two minimums

Answer 1

The correct option is A One maximum and one minimum

Given $f\left(x\right)=4x^5-25x^4+40x^3-10$

Differentiate on both sides with respective to $x$

$f^{\prime }\left(x\right)=20x^4-100x^3+120x^2$

$f^{\prime }\left(x\right)=x^2(20x^2-100x+120)$

$0=20x^2(x^2-5x+6)$

$0=20x^2(x-3)(x-2)$

$x=0$, $x=3$, $x=2$

Differentiate $f^{\prime }\left(x\right)$ on both sides with respective to $x$

$f^{\prime \prime }\left(x\right)=80x^3-300x^2+240x$

$f^{\prime \prime }\left(0\right)=0$

$f^{\prime \prime }\left(2\right)<0$

$f^{\prime \prime }\left(3\right)>0$

So their exist a minimum & a maximum