Question

The function $f\left(x\right)=2x^3-3x^2-12x+4$ has

• A. No maxima and minima
• B. One maximum and one minimum
• C. Two maxima
• D. Two minima

Answer 1

The correct option is B

One maximum and one minimum

Explanation for correct option

Step 1: Find the critical point

Given function $f\left(x\right)=2x^3-3x^2-12x+4$

$\Rightarrow f\text{'}\left(x\right)=6x^2-6x-12$

We know that point of maxima and minima occur at the critical point.

The critical point is given as $f\text{'}\left(x\right)=0$
$$\begin{gathered} \Rightarrow 6x^2-6x-12=0 \\ \Rightarrow x^2-x-2=0 \\ \Rightarrow (x+1)(x–2)=0 \end{gathered}$$

So $x=-1,2$

Step 2: Put the values of critical points to get maxima and minima

$$\begin{gathered} f\text{'}\left(x\right)=6x^2-6x-12 \\ f\text{'}\text{'}\left(x\right)=12x-6=6(2x-1) \end{gathered}$$

Now,

$$\begin{gathered} f\text{'}\text{'}(-1)=-18 \\ andf\text{'}\text{'}\left(2\right)=18 \end{gathered}$$

So, at $x=-1$,$f\left(x\right)$ has maxima and at $x=2$,$f\left(x\right)$ has minima.

Thus, $f\left(x\right)$ has one maxima and one minima.

Hence, the correct option is option(B)