f(x) is a cubic function with f(1)=-6, f(-1)=10 and has maxima at x=-1. If f^'(x) has minima at

step 1 question we have to find two be polynomial f x so let f x equal a x cube plus b x squared plus c

F(x) is a cubic function with f(1)=-6, f(-1)=10 and has...

$f(x)$ is a cubic function with $f(1)=-6, f(-1)=10$ and has maxima at $x=-1$. If $f^{\prime}(x)$ has minima at $x=1$, then
(A) $f(x)=x^{3}+3 x^{2}-9 x+5$
(B) $f(x)=x^{3}-3 x^{2}-9 x+5$
(C) $f(x)=x^{3}-3 x^{2}+9 x+5$
(D) $f(x)=x^{3}-3 x^{2}-9 x+5$

Key Concepts

Second Derivative Test

The second derivative of a function gives insight into its concavity. This test is used to confirm the nature of a critical point; for example, a negative second derivative at a point where the first derivative is zero indicates a local maximum, while a zero of the second derivative can indicate an inflection point or, in the context of a quadratic derivative, the location of a minimum or maximum.

Derivative and its Properties

The derivative of a function provides information about the rate of change and the slope of the function at any given point. For polynomial functions, taking the derivative reduces the degree of the polynomial, and the signs of the derivative indicate intervals of increase or decrease in the original function.

Polynomial Coefficient Determination

Determining the coefficients of a polynomial function involves setting up equations based on given function values and derivative conditions at specific points. By substituting the conditions into the polynomial and its derivatives, a system of equations is formed that can be solved to find the unknown coefficients.

Cubic Functions

Cubic functions are third?degree polynomials characterized by having at most two turning points and one point of inflection. They often exhibit an S-shaped curve and can have both a local maximum and a local minimum. Understanding the general structure of a cubic function is essential when determining its coefficients from given conditions.

Critical Points and Extremum Analysis

Critical points are the values of x where the first derivative of a function is zero or undefined, indicating potential local maximums or minimums. In the study of cubic functions, identifying these points helps in understanding the function’s behavior, such as when a function has a local maximum at a specific point.

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