Mathematics

Problem

Consider the cubic polynomial

f(x)=x^{3}-3kx^{2}+12x+5

, where

k

is a real parameter. a. (i) Obtain

f'(x)

and show that the locations of its critical points are the real roots of a quadratic equation in

x

that involves

k

. Hence write the discriminant of this quadratic in terms of

k

. b. (ii) Determine the range of

k

for which

f(x)

possesses two distinct real critical points, and classify each critical point as a local maximum or a local minimum. c. (iii) For one admissible value of

k

, verify your conclusion by sketching a rough graph of

f(x)

indicating the positions of the turning points and their nature.

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