Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

88_Extrema and the Geometric Applications of a Derivative [Ch. 8
greatest and least values of the function throughout the domain
of definition).
849. !/ = rih&. 853 - V = x* on the interval [ 1,3].
--
850. y = x(lOx). 854. y = 2x* + 3* 2
12* + 1
851. y= sin 4 A; + cos 4 A;. a) on the interval f 1,6];
b) on the interval [10,12],
852. # = arc cos x.
855. Show that for positive values of *we have the inequality
856. Determine the coefficients p and q of the quadratic trinomial
y*=x*+px + q so that this trinomial should have a min-
imum t/ = 3 when Jt= 1. Explain the result in geometrical terms.
857. Prove the inequality
Solution. Consider the function
e* > 1 + x when x 4* 0.
In the usual way we find lhat this function has a single minimum /(0)
Hence,
/(*)>/ (0) when x 0,
and so e* > 1 +x when x ^ 0,
as we set out to prove.
Prove the inequalities:
858. x ^< sin x < x when *>0.
o
859. cos*>l ^ when
860. A: ~