Consider a polynomial y=P(x) of the least degree passing through A(-1,1) and whose graph has two

step 1 So the question asks us, consider a polynomial y is equal to px, y is equal to p of x, y is equa

Consider a polynomial y=P(x) of the least degree passing...

Consider a polynomial $\mathrm{y}=\mathrm{P}(\mathrm{x})$ of the least degree passing through $\mathrm{A}(-1,1)$ and whose graph has two points of inflexion $\mathrm{B}(1,2)$ and $\mathrm{C}$ with abscissa $0 \mathrm{at}$ which the curve is inclined to the positive axis of abscissas at an angle of $\sec ^{-1} \sqrt{2}$. The value of $\mathrm{P}(-1)$ equals (A) $-1$ (B) 0 (C) 1 (D) 2

Consider a polynomial $\mathrm{y}=\mathrm{P}(\mathrm{x})$ of the least degree passing through $\mathrm{A}(-1,1)$ and whose graph has two points of inflexion $\mathrm{B}(1,2)$ and $\mathrm{C}$ with abscissa $0 \mathrm{at}$ which the curve is inclined to the positive axis of abscissas at an angle of $\sec ^{-1} \sqrt{2}$.
The value of $\mathrm{P}(-1)$ equals
(A) $-1$
(B) 0
(C) 1
(D) 2

Key Concepts

Polynomial Interpolation

Polynomial interpolation involves finding a polynomial that passes through a prescribed set of points. The polynomial is constructed by imposing constraints at given points, ensuring the function meets specific values. This concept forms the basis of understanding how to encode positional data into an algebraic expression.

Inflection Points

An inflection point is where a curve changes its concavity, meaning the second derivative of the function changes its sign. In the context of polynomials, the identification of inflection points adds extra derivative conditions (typically, that the second derivative equals zero) which are critical in determining the form of the polynomial.

Minimal Degree Polynomial

The concept of a minimal degree polynomial is about finding the polynomial of the smallest possible degree that satisfies all given conditions, such as passing through specific points and having designated behavior in terms of derivatives. This ensures that the polynomial is as simple as possible while still meeting the required specifications.

Tangent Lines and Slope

The slope of a tangent line to a curve at any given point is determined by the first derivative of the function at that point. Knowing the angle of inclination of the tangent to the x-axis gives a relation for the value of the first derivative, further constraining the polynomial.

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