MCQ
Choose the correct answer from the given four options : The function $f(x)=2 x^3-3 x^2-12 x+4,$ has :
- ATwo points of local maximum.
- BTwo points of local minimum.
- ✓One maxima and one minima.
- DNo maxima or minima.
✓
Answer
Correct option: C.
One maxima and one minima.
We have, $f(x)=2 x^3-3 x^2-12 x+4$
$\Rightarrow f^{\prime}(x)=6 x^2-6 x-12$
$\Rightarrow f^{\prime}(x)=6\left(x^2-x-2\right)$
$\Rightarrow f^{\prime}(x)=6(x+1)(x-2)$
Find the critical points by equating $f'(x)$ to $0.$
$\therefore f^{\prime}(x)=0$
$\Rightarrow 6(x+1)(x-2)=0$
$\Rightarrow x=-1$ and $x=+2$
From the above number line, we can conclude that, $x = -1$ is point of local maxima and $x = 2$ is point of local minima.
Thus, $f(x)$ has one maxima and one minima.
$\Rightarrow f^{\prime}(x)=6 x^2-6 x-12$
$\Rightarrow f^{\prime}(x)=6\left(x^2-x-2\right)$
$\Rightarrow f^{\prime}(x)=6(x+1)(x-2)$
Find the critical points by equating $f'(x)$ to $0.$
$\therefore f^{\prime}(x)=0$
$\Rightarrow 6(x+1)(x-2)=0$
$\Rightarrow x=-1$ and $x=+2$
From the above number line, we can conclude that, $x = -1$ is point of local maxima and $x = 2$ is point of local minima.
Thus, $f(x)$ has one maxima and one minima.
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