Number of roots of the function f( x )=(1/( x +1)3)-3 x + sin x is

Question

Number of roots of the function f( x )=(1/( x +1)3)-3 x + sin x is (A) 0 (B) 1 (C) 2 (D) more than 2

Tardigrade Admin · Accepted Answer

f prime(x)=-(3/(x+1)4)-3+ cos x<0 hence f(x) is always decreasing, Also as x arrow ∞, f(x) arrow-∞ and as x arrow-∞, f(x) arrow+∞ hence one positive and one negative root Graph is as shown

Q. Number of roots of the function $f (x) = \frac{1}{( x + 1 ) ^{3}} - 3 x + sin x$ is

0

1

2

more than 2

$f^{'} (x) = - \frac{3}{( x + 1 ) ^{4}} - 3 + cos x < 0$
hence $f (x)$ is always decreasing, Also as $x \to \infty, f (x) \to - \infty$ and as $x \to - \infty, f (x) \to + \infty$ hence one positive and one negative root
Graph is as shown

Q. Number of roots of the function f(x)=(x+1)31​−3x+sinx is

0

1

2

more than 2

f′(x)=−(x+1)43​−3+cosx<0 hence f(x) is always decreasing, Also as x→∞,f(x)→−∞ and as x→−∞,f(x)→+∞ hence one positive and one negative root Graph is as shown

Q. Number of roots of the function $f (x) = \frac{1}{( x + 1 ) ^{3}} - 3 x + sin x$ is

$f^{'} (x) = - \frac{3}{( x + 1 ) ^{4}} - 3 + cos x < 0$
hence $f (x)$ is always decreasing, Also as $x \to \infty, f (x) \to - \infty$ and as $x \to - \infty, f (x) \to + \infty$ hence one positive and one negative root
Graph is as shown