The function f(x)=\;{4x³-3x²}{6}-2 x+(2x-1) x [24 Feb 2021 Shift 1]

Correct Answer is : increases in [ 12,∞)[\;1/2, ∞)[21,∞)

Correct Answer is : increases in [ 12,∞)[\;1/2, ∞)[21,∞)

Test Index

Application of Derivatives Part 2

Section: Mathematics

Question : 99 of 100

Marks: +1, -0

The function

f(x)=\;\frac{4x^{3}-3x^{2}}{6}-2\sin x+(2x-1)\cos x

Solution:

Given,

f(x)=\;\frac{4x^{3}-3x^{2}}{6}-2\sin x+(2x-1)\cos x

f'(x)=\;\frac{12x^{2}-6x}{6}-2\cos x+(2x-1)(-\sin x)+\cos x(2)

=(2x^{2}-x)-2\cos x-2x\sin x+\sin x+2\cos x

=2x^{2}-x-2x\sin x+\sin x

=2x(x-\sin x)-1(x-\sin x)

f'(x)=(2x-1)(x-\sin x)

for

x > 0, x-\sin x > 0

x < 0, x-\sin x < 0

for

x \in \left(-\infty, 0] \cup [\;\frac{1}{2}, \infty\right), f'(x) \ge 0

for

x \in \left[0, \;\frac{1}{2}\right], f'(x) \le 0

Hence,

f(x)

increases in

[\;\frac{1}{2}, \infty)

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