Question

Let $f(x)=2 x+\tan ^{-1} x$ and $g(x)=\log _e\left(\sqrt{1+x^2}+x\right), x \in[0,3]$ Then

• A. $\min f^{\prime}(x)=1+\max g^{\prime}(x)$
• B. there exist $0 < x_1 < x_2 < 3$ such that $f(x) < g(x), \forall x \in\left(x_1, x_2\right)$
• C. $\max f(x)>\max g(x)$
• D. there exists $\hat{x} \in[0,3]$ such that $f^{\prime}(\hat{x}) < g^{\prime}(\hat{x})$

Answer 1

Answer: (C)

$\max f(x)>\max g(x)$

Answer 2

The correct answer is (C) : $\max f(x)>\max g(x)$
$f'(x)=2+\frac{1}{1+x^2},\ g'(x)=\frac{1}{\sqrt{1+x^2}}$
Both does not have critical values
$f(0)=0,f(3)=6+\tan^{-1}(3)$
$g(0)=0,g(3)=\log(3+\sqrt{10})$
Let h(x) = f(x) - g(x)
$h'(x) > 0∀x∈(0,3)$
∴ h(x) is increasing function