For the quadratic function

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

p(x)=x^{2}+2(k-1)x+2k+5

, where

k\in\mathbb{R}

, answer the following: (i) Obtain the coordinates of the vertex in terms of

k

. (ii) Hence find an expression for the minimum value of

p(x)

in terms of

k

. (iii) Determine all real values of

k

for which the minimum value of

p(x)

is exactly

4

. (iv) For each value of

k

obtained in part (iii), calculate the corresponding value of

x

at which

p(x)

attains its minimum.