Assertion (A) and Reason (R) (A) Both A and R are true and R is the correct explanation of A. (B

step 1 So in this question we have f of x is equal to 3x dash to the power 4 minus 4 x cube plus 6 x sq

Assertion (A) and Reason (R) (A) Both A and R are true and R...

Assertion (A) and Reason (R) (A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$. (B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$. (C) $\mathrm{A}$ is true, $\mathrm{R}$ is false. (D) $\mathrm{A}$ is false, $\mathrm{R}$ is true. Assertion $(\mathrm{A}): \mathrm{A}$ tangent parallel to $\mathrm{x}$-axis can be drawn for $f(x)=(x-1)(x-2)(x-3)$ in the interval $[1,3]$ Reason (R): A horizontal tangent can be drawn to any cubic function.

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(\mathrm{A}): \mathrm{A}$ tangent parallel to $\mathrm{x}$-axis can be drawn for $f(x)=(x-1)(x-2)(x-3)$ in the interval $[1,3]$ Reason (R): A horizontal tangent can be drawn to any cubic function.

Key Concepts

Tangent to a Curve

A tangent to a curve is a straight line that touches the curve at a single point without crossing it at that point, representing the instantaneous direction of the curve. It is a fundamental concept in calculus and geometry that reflects the local linear approximation of the function at the point of contact.

Horizontal Tangent

A horizontal tangent is a tangent line that is parallel to the x-axis, which implies that the slope of the tangent is zero. This condition occurs when the derivative of the function at the point of tangency is equal to zero, indicating either a local maximum, a local minimum, or a saddle point in the function.

Derivative and Critical Points

The derivative of a function at a particular point represents the slope of the tangent line at that point. Setting the derivative equal to zero helps identify critical points, where the function may have a horizontal tangent. These points are important for analyzing the behavior of functions, such as determining relative extrema and inflection points.

Properties of Cubic Functions

Cubic functions are polynomial functions of degree three that can exhibit a variety of behaviors including inflection points and local extrema. While cubic functions may have horizontal tangents if their derivative equals zero at some points, it is not guaranteed that every cubic function will have one, as some may be monotonic over their entire domain.

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