Find the range of the function f:[0, 1]→ R, f(x)=x3-x2+4x+2 sin -1x ?

Question

Find the range of the function f:[0, 1]→ R, f(x)=x3-x2+4x+2 sin -1x ? (A)  [-(π +2), 0]  (B)  [0,4+π ]  (C)  [2,3]  (D)  (0, 2+π ]

Tardigrade Admin · Accepted Answer

We have f:[0, 1]→ R f(x)=x3-x2+4x+2 sin -1x Now f'(x)=3x2-2x+4+(2/√1-x2) For x [0, 1], f' (x)>0 Hence, it is a increasing function at x=0, f(0)=0-0+4(0)+2sin-1(0)=0 at x=1,f(1)=1-1+4(1)+2 sin -1(1) =4+2( (π /2) )=4+π ∴ Range of f(x) [0, 4+π ]

Q. Find the range of the function $f : [0, 1] \to R, f (x) = x^{3} - x^{2} + 4 x + 2 sin^{- 1} x$ ?

$[- (π + 2), 0]$

$[0, 4 + π]$

$[2, 3]$

$(0, 2 + π]$

Q. Find the range of the function f:[0,1]→R,f(x)=x3−x2+4x+2sin−1x ?

[−(π+2),0]

[0,4+π]

[2,3]

(0,2+π]

We have f:[0,1]→R f(x)=x3−x2+4x+2sin−1x Now f′(x)=3x2−2x+4+1−x2​2​ For x[0,1],f′(x)>0 Hence, it is a increasing function at x=0,f(0)=0−0+4(0)+2sin−1(0)=0 at x=1,f(1)=1−1+4(1)+2sin−1(1) =4+2(2π​)=4+π ∴ Range of f(x)[0,4+π]

Q. Find the range of the function $f : [0, 1] \to R, f (x) = x^{3} - x^{2} + 4 x + 2 sin^{- 1} x$ ?

$[- (π + 2), 0]$

$[0, 4 + π]$

$[2, 3]$

$(0, 2 + π]$