Question

$f\left(x\right)=x^5-5x^4+5x^3+1$ has

• A. Two maximum and two minimum value
• B. Two maximum and one minimum value
• C. One maximum and one minimum value
• D. None of the above

Answer 1

The correct option is C

One maximum and one minimum value

Explanation for the correct option:

Step 1. Find the extrema of the given function:

$f\left(x\right)=x^5–5x^4+5x^3+1$

$f\text{'}\left(x\right)=5x^4–20x^3+15x^2$

$f\text{'}\text{'}\left(x\right)=20x^3–60x^2+30x$

For maxima and minima, put $f\text{'}\left(x\right)=0$

⇒ $5x^2(x^2–4x+3)=0$

⇒ $5x^2(x–3)(x–1)=0$

⇒ $x=0,1,3$

Step 2. Put the values of $x$ in $f\text{'}\left(x\right)$ and $f\text{'}\text{'}\left(x\right)$

$f\text{'}{\left(x\right)}_{x=0}=0$

$f\text{'}\text{'}{\left(x\right)}_{x=1}=20–60+30$

$=–10<0$

$f\text{'}\text{'}{\left(x\right)}_{x=3}=20\times 27–60\times 9+30\times 3$

$=540–540+90>0$

$\therefore$$f\left(x\right)$ is maxima at $x=1$ and maximum value is $f\left(x\right)=2$ and $f\left(x\right)$ is minima at $x=3$ and minimum value $f\left(x\right)=–26$

Thus, $f\left(x\right)$ has $1$ maximum and $1$ minimum value.

Hence, Option ‘C’ is Correct.