Let f(x)=(x-4)(x-5)(x-6)(x-7) then (a) f^'(x)=0 has four real roots (b) three roots of f^'(x)=0

step 1 in this problem since we have f4 is equals to f5 is equals to f6 which is equals to f7 which is

Let f(x)=(x-4)(x-5)(x-6)(x-7) then (a) f^'(x)=0 has four...

Let $f(x)=(x-4)(x-5)(x-6)(x-7)$ then
(a) $f^{\prime}(x)=0$ has four real roots
(b) three roots of $f^{\prime}(x)=0$ lie in $(4,5) \cup(5,6)$ $\cup(6,7)$
(c) the equation $f^{\prime}(x)=$ has only two roots
(d) three roots of $f^{\prime}(x)=0$ lie in $(3,4) \cup(4,5)$ $\cup(5,6)$

Key Concepts

Rolle's Theorem

Rolle’s Theorem states that if a continuous function is differentiable on an open interval and has equal values at two distinct endpoints, then there must be at least one point between where the derivative is zero. This concept is essential when deducing that between every pair of consecutive roots of a polynomial, there exists at least one root of its derivative.

Critical Points and Derivatives

Critical points of a polynomial occur where its derivative is zero. Identifying these points reveals where the function achieves local extrema or changes its curvature. Analyzing the number and location of these points is crucial for understanding the overall behavior of the function.

Degree of a Polynomial and Its Derivative

The degree of a polynomial governs the maximum number of its roots, and similarly, the derivative of an n-degree polynomial is an (n-1)-degree polynomial. This relation provides an upper limit to the number of real roots (critical points) that the derivative can have and is fundamental when analyzing the function’s structure.

Distribution of Roots

The placement of the roots of a polynomial influences where the roots of its derivative must lie. This idea, combined with Rolle’s Theorem, leads to the conclusion that between any two consecutive roots of the original polynomial there is at least one root of the derivative, which helps in assessing statements about the locations of these critical points.

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