3.2 Continuity, Derivatives, and Integrals 141Below you will find a variety of problems based on the above-mentioned theorems
(Rolle, Lagrange, Cauchy). Try to solve them, remembering that “good judgment comes
from experience, and experience comes from bad judgment’’ (Barry LePatner).
415.Prove that not all zeros of the polynomialP(x)=x^4 −
√
7 x^3 + 4 x^2 −√
22 x+ 15
are real.416.Letf:[a, b]→Rbe a function, continuous on[a, b]and differentiable on(a, b).
Prove that if there existsc∈(a, b)such that
f(b)−f(c)
f(c)−f(a)
< 0 ,
then there existsξ∈(a, b)such thatf′(ξ )=0.417.Forx≥2 prove the inequality
(x+ 1 )cos
π
x+ 1−xcos
π
x> 1.
418.Letn>1 be an integer, and letf:[a, b]→Rbe a continuous function,n-times
differentiable on(a, b), with the property that the graph offhasn+1 collinear
points. Prove that there exists a pointc∈(a, b)with the property thatf(n)(c)=0.
419.Letf:[a, b]→Rbe a function, continuous on[a, b]and differentiable on(a, b).
LetM(α, β)be a point on the line passing through the points(a, f (a))and(b, f (b))
withα/∈[a, b]. Prove that there exists a line passing throughMthat is tangent to
the graph off.
420.Letf:[a, b]→Rbe a function, continuous on[a, b]and twice differentiable on
(a, b).Iff(a)=f(b)andf′(a)=f′(b), prove that for every real numberλthe
equation
f′′(x)−λ(f′(x))^2 = 0
has at least one solution in the interval(a, b).
421.Prove that there are no positive numbersxandysuch that
x 2 y+y 2 −x=x+y.422.Letαbe a real number such thatnαis an integer for every positive integern. Prove
thatαis a nonnegative integer.
423.Find all real solutions to the equation
6 x+ 1 = 8 x− 27 x−^1.424.LetP(x)be a polynomial with real coefficients such that for every positive integer
n, the equationP(x)=nhas at least one rational root. Prove thatP(x)=ax+b
withaandbrational numbers.