Question

Let $f$ be the function for $f\left(x\right)=\cos x-(1-\frac{x^2}{2})$, then $f\left(x\right)$ is strictly increasing in the interval

• A. $(-\infty ,\infty )$
• B. $(-2,\infty )$
• C. $[0,\infty )$
• D. $(0,\infty )$

Answer 1

Answer: (D)

$(0,\infty )$

Given, $f\left(x\right)=\cos x-(1-\frac{x^2}{2})$

then, $f^{{}^{\prime }}\left(x\right)=-\sin x-(-\frac{2x}{2})f^{{}^{\prime }}\left(x\right)=x-\sin x$

$\forall x>0x>\sin x$

$f^{{}^{\prime }}\left(x\right)>0\forall x>0$

$f^{{}^{\prime }}\left(x\right)<0\forall x<0$

So, f(x) is strictly increasing in the interval $(0,\infty ).$