Amit M Agarwal Solutions for Chapter: dy/dx as a Rate Measurer & Tangents, Normals, Exercise 3: Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1

If $f$ is an odd continuous function in $\left[-\mathrm{1,} 1\right]$ and differentiable in $\left(-\mathrm{1,} 1\right)$, then:

The parabola $y=x^2+px+q$ intercepts the straight line $y=2x-3$ at a point with abscissa $1.$ If the distance between the vertex of the parabola and the $x-$axis is least, then:

The abscissa of the point on the curve $\sqrt{xy}=a+x$, the tangent at which cuts off equal intercepts from the coordinate axes is $\left(a>0\right)$:

If the side of a triangle vary slightly $\left(da, db, dc\right)$ in such a way that its circumradius remains constant, then $\frac{da}{\cos A}+\frac{db}{\cos B}+\frac{dc}{\cos C}$ is equal to

For function $f\left(x\right)=\frac{\mathrm{l}\mathrm{n}x}{x}$, which of the following statements are true?

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Statement I: $f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{2}-x,& x<\frac{1}{2}\\ {\left(\frac{1}{2}-x\right)}^2,& x\ge \frac{1}{2}\end{array}\right.$. Mean value theorem is applicable in the interval $\left[\mathrm{0,} 1\right]$
Statement II: For the application of mean value theorem, $f\left(x\right)$ must be continuous
in $\left[0, 1\right]$ and differentiable in $\left(0, 1\right)$.

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Let $f\left(x\right)=\ln \left(2+x\right)-\frac{2x+2}{x+3}$
Statement I: The function $f\left(x\right)=0$ has a unique solution in the domain of $f\left(x\right)$.
Statement II: If $f\left(x\right)$ is continuous in $\left[a, b\right]$ and is strictly monotonic in $\left(a, b\right)$,
then $f$ has a unique root in $\left(a, b\right)$.

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Consider the polynomial function
$f\left(x\right)=\frac{x^7}{7}-\frac{x^6}{6}+\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x$
Statement I: The equation $f\left(x\right)=0$ cannot have two or more roots.
Statement II: Rolle's theorem is not applicable for $y=f\left(x\right)$ on any interval
$\left[a, b\right]$, where $a, b\in R$.