Question

If at each point of the curve$y=x^3-a{x}^2+x+1$ tangent is inclined at an acute angle with the positive direction of the X-axis, then

• A. $a\le \surd 3$
• B. $a>0$
• C. $–\surd 3<a<\surd 3$
• D. None of these

Answer 1

The correct option is C

$–\surd 3<a<\surd 3$

The explanation of the correct option :

Step1. The given curve :

The given curve $y=x^3-a{x}^2+x+1$

Slope of the tangent $=\frac{dy}{dx}=\tan$

$\frac{dy}{dx}>0$

$$\begin{gathered} y=x^3–a{x}^2+x+1 \\ \Rightarrow \frac{dy}{dx}=3x^2–2ax+1 \\ \Rightarrow 3x^2–2ax+1>0 \end{gathered}$$

Step2. To find the range of $\mathit{a}$:

Now, discriminant $\Delta <0$

$$\begin{gathered} \Rightarrow b^2-4ac<0 \\ \Rightarrow {(-2a)}^2–4\times 3\times 1<0 \\ \Rightarrow 4a^2–12<0 \\ \Rightarrow a^2<3 \end{gathered}$$

Thus $–\surd 3<a<\surd 3$

Hence the correct option is C.