(A) The intercept of the common tangent to the curves (P) -1...

(A) The intercept of the common tangent to the curves (P) $-1$ $y^{2}=8 x$ and $x y=-1$ on the axis of $y$ is equal to (B) Let $f$ be a real function whose derivates upto third order exist and for (Q) 0 some pair $a, b \in R, a<b \log \frac{f(a)+f^{\prime}(a)+f^{\prime \prime}(a)}{\left. \left.f(b)+f^{\prime}(b)+f^{\prime \prime}\right) b\right)}=a-b$, then there exists $\mathrm{c}(\mathrm{a}, \in \mathrm{b})$ for which $\frac{\mathrm{f}^{\prime \prime}(\mathrm{c})}{\mathrm{f}(\mathrm{c})}$ is equal to (C) Let $f(x)=\left(x^{2}-1\right)\left(x^{2}-4\right)$, and $\alpha, \beta, \gamma$ be the roots of the equation (R) 1 $f^{\prime}(x)=0$ then $[\alpha]+[\beta]+[\gamma]$ is equal to $([t]$ represents the integral part of 1$)$ (D) If three normals can be drawn to the curve $y^{2}=x$ (S) $5 / 4$ from the point $(\mathrm{c}, 0)$ then $\mathrm{c}$ can be equal to $\quad$ (T) 2

(A) The intercept of the common tangent to the curves
(P) $-1$ $y^{2}=8 x$ and $x y=-1$ on the axis of $y$ is equal to
(B) Let $f$ be a real function whose derivates upto third order exist and for
(Q) 0 some pair $a, b \in R, a<b \log \frac{f(a)+f^{\prime}(a)+f^{\prime \prime}(a)}{\left. \left.f(b)+f^{\prime}(b)+f^{\prime \prime}\right) b\right)}=a-b$, then there
exists $\mathrm{c}(\mathrm{a}, \in \mathrm{b})$ for which $\frac{\mathrm{f}^{\prime \prime}(\mathrm{c})}{\mathrm{f}(\mathrm{c})}$ is equal to
(C) Let $f(x)=\left(x^{2}-1\right)\left(x^{2}-4\right)$, and $\alpha, \beta, \gamma$ be the roots of the equation
(R) 1 $f^{\prime}(x)=0$ then $[\alpha]+[\beta]+[\gamma]$ is equal to $([t]$ represents the integral part of 1$)$
(D) If three normals can be drawn to the curve $y^{2}=x$
(S) $5 / 4$ from the point $(\mathrm{c}, 0)$ then $\mathrm{c}$ can be equal to $\quad$ (T) 2

Key Concepts

Tangent Lines

Tangent lines to a curve represent the instantaneous direction in which the curve is heading, and their equations can be determined using differentiation. In problems involving common tangents, the line must satisfy the tangency condition for more than one curve, meaning its slope and point of contact must be appropriately determined by setting up equations involving derivatives.

Normal Lines

Normal lines to a curve are perpendicular to the tangent at the point of contact. They are crucial when studying the geometry of curves because they often provide information about symmetry, distance, or extremal properties. Problems involving normals require understanding the relationship between the slope of the tangent and that of the normal, typically through the negative reciprocal.

Mean Value Theorem (and Rolle's Theorem)

The Mean Value Theorem is a fundamental result in calculus stating that for functions continuous on a closed interval and differentiable on the open interval, there exists a point where the instantaneous rate of change equals the average rate of change over the interval. A variant, Rolle’s Theorem, is commonly used to establish the existence of roots of derivatives under certain conditions. These theorems are essential in problems that require proving the existence of a point where a particular derivative property holds.

Higher Order Derivatives

Analyzing higher order derivatives (such as second and third derivatives) gives deeper insights into the behavior of functions, including concavity, inflection points, and the approximation of functions. In applied problems, showing the relationship between a function and its higher derivatives can lead to conditions or equations that guarantee certain geometric or analytic properties.

Polynomial Differentiation and Critical Points

For polynomial functions, differentiation is straightforward and allows for the determination of critical points where the derivative vanishes. These points often correspond to local extrema or points of inflection, and analyzing them can involve techniques such as factoring and applying the Fundamental Theorem of Algebra to count and identify the properties of the roots.

Intercept Calculations

Intercept calculations involve finding the points where a line or curve crosses the axes, typically by setting the corresponding variable (x or y) to zero. In geometric problems, particularly those involving tangents or normals, computing intercepts is essential for understanding the intersection of lines with coordinate axes and helps in determining specific attributes of the line such as its distance from the origin.

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