Find the absolute maximum and absolute minimum values of f on the given interval.

$f\left(x\right)=2{x}^{3}-3{x}^{2}-12x+1,[-2,3]$

Yolanda Jorge
2021-12-07
Answered

Find the absolute maximum and absolute minimum values of f on the given interval.

$f\left(x\right)=2{x}^{3}-3{x}^{2}-12x+1,[-2,3]$

You can still ask an expert for help

Rosemary McBride

Answered 2021-12-08
Author has **10** answers

Step 1

The function is given by

$f\left(x\right)=2{x}^{3}-3{x}^{2}-12x+1,[-2,3]$

To evaluate : The absolute maximum and minimum of the function on the given interval.

Step 2

Let us evaluate the critical numbers,

$\frac{df}{dx}=0$

$\Rightarrow \frac{d}{dx}(2{x}^{3}-3{x}^{2}-12x+1)=0$

$\Rightarrow 6{x}^{2}-6x-12=0$

$\Rightarrow {x}^{2}-x-2=0$

$\Rightarrow {x}^{2}-2x+x-2=0$

$\Rightarrow x(x-2)+1(x-2)=0$

$\Rightarrow (x+1)(x-2)=0$

$\Rightarrow x=-1,2$ These are critical numbers

Step 3

Now, let us evaluate the values of the function at the critical numbers and at the endpoints of the closed interval [-2,3]

At x=-1,

$f(-1)=2\times {(-1)}^{3}-3\times {(-1)}^{2}-12\times (-1)+1=8$

Absolute maximum

At x=2,

$f\left(2\right)=2\times {\left(2\right)}^{3}-3\times {\left(2\right)}^{2}-12\times \left(2\right)+1=-19$

Absolute minimum

At x=-2,

$f(-2)=2\times {(-2)}^{3}-3\times {(-2)}^{2}-12\times (-2)+1=-3$

At x=3,

$f\left(3\right)=2\times {\left(3\right)}^{3}-3\times {\left(3\right)}^{2}-12\times \left(3\right)+1=-8$

Step 4

Hence, the absolute maximum and absolute minimum values of the function are :

Absolute maximum value of the function = 8

Absolute minimum value of the function =-19

The function is given by

To evaluate : The absolute maximum and minimum of the function on the given interval.

Step 2

Let us evaluate the critical numbers,

Step 3

Now, let us evaluate the values of the function at the critical numbers and at the endpoints of the closed interval [-2,3]

At x=-1,

Absolute maximum

At x=2,

Absolute minimum

At x=-2,

At x=3,

Step 4

Hence, the absolute maximum and absolute minimum values of the function are :

Absolute maximum value of the function = 8

Absolute minimum value of the function =-19

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