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 (A) : For all a, b $\in \mathrm{R}$, the function $f(x)=3 x^{4}-$ $4 x^{3}+6 x^{2}+a x+b$ has exactly one extremum. Reason $(\mathbf{R}):$ If a cubic function is monotonic, then its graph cuts $\mathrm{x}$-axis only once.

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 (A) : For all a, b $\in \mathrm{R}$, the function $f(x)=3 x^{4}-$ $4 x^{3}+6 x^{2}+a x+b$ has exactly one extremum. Reason $(\mathbf{R}):$ If a cubic function is monotonic, then its graph cuts $\mathrm{x}$-axis only once.

Key Concepts

Polynomials and Their Derivatives

Polynomials, due to their continuous and differentiable nature, are often analyzed by studying their derivatives. The degree of the polynomial directly influences the behavior and number of its critical points. Understanding how the derivative behaves provides insights into the overall shape of the polynomial, including the number and types of its extrema.

Monotonic Cubic Functions

A cubic function is a third-degree polynomial whose graph can display various behaviors depending on its coefficients. When a cubic function is monotonic, it means that its rate of change does not reverse sign, leading to a consistent increasing or decreasing behavior. This monotonicity ensures that the function has a single real root, often utilized in explaining why a related polynomial intersects an axis only once.

Critical Points

Critical points are the values in the domain of a function where its derivative is either zero or undefined. These points are important because they are candidates for local maxima or minima, i.e., the extreme values of the function. In the context of polynomial functions, the smoothness of the function ensures that the derivative's zeros signal where the function's behavior may change from increasing to decreasing or vice versa.

Monotonicity

Monotonicity refers to a function consistently increasing or decreasing throughout its domain without any changes in direction. A monotonic function has no local extrema (except possibly at the boundaries) because it does not reverse direction. This concept is central when analyzing functions because a monotonic derivative implies a predictable, one-time behavior in crossing specific values, such as the x-axis.

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