If f:[2,3] arrow R is defined by f(x)=x3+3 x-2, then the range of f(x) is contained in the interval

Question

If f:[2,3] arrow R is defined by f(x)=x3+3 x-2, then the range of f(x) is contained in the interval (A) [1,12] (B)  [12,34] (C) [35,50] (D)  [-12,12]

Tardigrade Admin · Accepted Answer

Given, f(x)=x3+3 x-2 On differentiating w.r.t. x, we get f prime(x)=3 x2+3>0 ∀ x ∈ R ∴ f(x) is increasing At x=2, f(2)=23+3(2)-2=12 At x=3, f(3)=33+3(3)-2=34 ∴ f(x) ∈[12,34]

Q. If $f : [2, 3] \to R$ is defined by $f (x) = x^{3} + 3 x - 2$ , then the range of $f (x)$ is contained in the interval

$[1, 12]$

$[12, 34]$

$[35, 50]$

$[- 12, 12]$

Given, $f (x) = x^{3} + 3 x - 2$
On differentiating w.r.t. $x$ , we get
$f^{'} (x) = 3 x^{2} + 3 > 0\forall x \in R$
$∴ f (x)$ is increasing
At $x = 2, f (2) = 2^{3} + 3 (2) - 2 = 12$
At $x = 3, f (3) = 3^{3} + 3 (3) - 2 = 34$
$∴ f (x) \in [12, 34]$

Q. If f:[2,3]→R is defined by f(x)=x3+3x−2, then the range of f(x) is contained in the interval

[1,12]

[12,34]

[35,50]

[−12,12]