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Chapter 6: Problem 81
\(h(x)=2(1+\sin 4 x)=4\) \(\Rightarrow \sin 4 \mathrm{x}=1\) \(\Rightarrow 4 \mathrm{x}=(4 \mathrm{n}+1) \frac{\pi}{2}\) \(x=\frac{(4 n+1) \pi}{8}\) Hence, A is correct
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A) \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\frac{4}{\mathrm{x}^{2}}+7\) \(f^{\prime}(x)=3 x^{2}-\frac{8}{x^{3}}=\frac{3 x^{5}-8}{x^{3}}\) \(\mathrm{f}(\mathrm{x})\) is \(\uparrow\) in \((-\infty, \infty)\) B) \(g(x)=\sqrt{t}+\sqrt{1+t}-4\) \(g^{\prime}(x)=\frac{1}{2 \sqrt{t}}+\frac{1}{2 \sqrt{1+t}}>0\) \(\mathrm{g}(\mathrm{x})\) is increasing in \((0, \infty)\) C) \(\mathrm{r}(\theta)=\theta+\sin ^{2}\left(\frac{\theta}{3}\right)-8\) \(r^{\prime}(\theta)=1+\frac{1}{3} \sin \left(\frac{2 \theta}{3}\right)>0\) \(\mathrm{r}(\theta)\) is an increasing function D) \(r(\theta)=\tan \theta-\cot \theta-\theta\) \(r^{\prime}(\theta)=\sec ^{2} \theta+\operatorname{cosec}^{2} \theta-1\) \(=\tan ^{2} \theta+\operatorname{cosec}^{2} \theta\) \(\mathrm{r}(\theta)\) is an increasing function Hence, $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ is correct
\(f(x)=x^{3}+2 x^{2}+5 x+2 \cos x\) \(f^{\prime}(x)=3 x^{2}+4 x+5-2 \sin x\) Min value of \(3 \mathrm{x}^{2}+4 \mathrm{x}+5\) is \(\frac{22}{3}\) \(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})>0\) \(f(0)=2 \& f(\infty)=\infty, \quad f(-\infty)-\infty\) There is no root in \([0,2 \pi]\) Hence, D is correct
As \(f(x)\) is increasing function \(\Rightarrow 2 a^{2}+a+1>3 a^{2}-4 a+1\) \(a^{2}-5 a<0\) \(a(a-5)<0\) as \(x \in(0, \infty), \quad a \in\\{2,3,4\\}\) Hence, \(\mathrm{B}, \mathrm{C}, \mathrm{D}\) is correct
A: $\begin{aligned} f(x) &=x^{4}-8 x^{3}+22 x^{2}-24 x+21 \\ f^{\prime}(x) &=4 x^{3}-24 x^{2}+44 x-24 \\ &=4\left(x^{3}-6 x^{2}+11 x-6\right) \\\ &=4(x-1)\left(x^{2}-5 x+6\right) \\ &=4(x-1)(x-2)(x-3) \end{aligned}$
Similar to ques 40 . \(f(x)=a x^{3}+b x^{2}+c x+d\)
\(f^{\prime}(x)=3 a x^{2}+2 b x+c>0\)
\(\Rightarrow 4 b^{2}-12 a c<0 \& a>0\)
\(\Rightarrow \mathrm{b}^{2}-3 \mathrm{ac}<0\)
\(b^{2}<3 a c \& a>0,3 b^{2}
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