Mathematics

Problem

Consider the family of functions

h_k(x)=\sin x+kx

on the interval

[0,\pi]

, where

k

satisfies

0\le k\le 2

. Which of the following statements best describes how the location of the absolute maximum of

h_k

changes as

k

increases? a) The maximizing

x

is always

x=\dfrac{\pi}{2}

for every

k

in the given range. b) The maximizing

x

moves rightward from

x=\dfrac{\pi}{2}

toward

x=\pi

k

increases from

0

1

and remains fixed at

x=\pi

for

k>1

. c) The maximizing

x

moves leftward from

x=\pi

toward

x=\dfrac{\pi}{2}

k

increases, reaching

x=\dfrac{\pi}{2}

when

k=2

. d) The maximizing

x

is an interior critical point for all

k\in[0,2]

and therefore never occurs at an endpoint.