Let a ∈R and let f: R →R be given by f(x)=x^5-5 x+a. Then (a) f(x) has three real roots if a>4 (

step 1 Let F of X equal to X raised to 5 minus 5x plus A. Then F of X equal to 0 implies X raised to 5

Let a ∈R and let f: R →R be given by f(x)=x^5-5 x+a. Then...

Let $a \in R$ and let $f: R \rightarrow R$ be given by $f(x)=x^{5}-5 x+a$. Then
(a) $f(x)$ has three real roots if $a>4$
(b) $f(x)$ has only real root if $a>4$
(c) $f(x)$ has three real roots if $a<-4$
(d) $f(x)$ has three real roots if $-4<a<4$

Key Concepts

Critical Points and Derivative Analysis

By taking the derivative of a polynomial, one can find its critical points, which are the potential locations for local maxima and minima. These points are particularly important in determining the number of real roots, as they indicate where a function changes direction. Analyzing the value of the function at these critical points helps in understanding how the parameters affect the crossing of the horizontal axis.

Polynomial Function Behavior

This concept involves understanding how a polynomial function behaves as its input values change. In particular, for functions of odd degree with a positive leading coefficient, the function will tend to negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity. This behavior guarantees at least one real root according to the Intermediate Value Theorem, as the function must cross the horizontal axis somewhere between these extremes.

Intermediate Value Theorem

This theorem states that if a continuous function has values of opposite signs at two points, then it must cross zero at some point between them. For polynomial functions, which are continuous everywhere, this theorem is crucial for proving the existence of real roots, particularly when the function's behavior at the extremes of the input domain is known.

Effect of Parameter Shifts on Roots

Adding a constant term to a polynomial function shifts its graph vertically. This parameter shift can alter the number and nature of the real roots by affecting whether the function's local extrema are above or below the x-axis. Understanding these effects is key when analyzing families of polynomials that vary by a parameter, as it determines the conditions under which multiple or unique roots occur.

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