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CGP EDU Academic Team
Published on: August 13, 2026
Find the values of the parameter ‘k’ for which the equation x 4 + 4x 3 – 8x 2 + k = 0 has all roots real.
Text Solution
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CHECK THE SOLUTION.
(k ∈ [0,3])
Let f(x) = x 4 + 4x 3 – 8x 2 + k
f’(x) = 4x 3 + 12x 2 – 16 x
= 4x (x 2 + 3x – 4) = 4x (x + 4) (x – 1)
⇒ f’(x) = 0 ⇒ x = – 4 , 0, 1
f’’ (x) = 12x 2 + 24x – 16 = 4(3x 2 + 6x – 4)
f’’ (–4) = 20 > 0
f’’(0) = – 16 < 0
f’’(1) = 20 > 0
⇒ x = – 4 and x = 1 are points of local minima whereas
x = 0 is point of local maxima
for f(x) = 0 to have 4 real roots
f(–4) < 0 ⇒ k < 128
f(0) > 0 ⇒ k > 0
f(1) < 0 ⇒ k < 3 ⇒ k ∈ (0, 3)
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