Mathematics

Mathematics

Problem

For a real parameter

k

, define the function

g_k:[0,\tfrac{\pi}{2}]\to\mathbb{R}

by

g_k(x)=k\sin x-\cos x

. a. (i) Compute

g_k'(x)

. b. (ii) Determine all real values of

k

for which

g_k

is increasing on the entire interval

\bigl[0,\tfrac{\pi}{2}\bigr]

. c. (iii) For the least value of

k

obtained in (ii), evaluate

g_k(0)

.

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