What will be the maxima for the function f(x) = x^4 –8x^3 + 22x^2 –24x + 8?

Category: QuestionsWhat will be the maxima for the function f(x) = x^4 –8x^3 + 22x^2 –24x + 8?
Editor">Editor Staff asked 11 months ago

What will be the maxima for the function f(x) = x^4 –8x^3 + 22x^2 –24x + 8?
 
(a) 0
 
(b) 1
 
(c) 2
 
(d) 3
 
This question was addressed to me in my homework.
 
This interesting question is from Calculus Application topic in division Application of Calculus of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

Right choice is (c) 2
 
Easiest explanation: We have, x^4 –8x^3 + 22x^2 –24x + 8 ……….(1)
 
Differentiating both sides of (1) with respect to x, we get,
 
f’(x) = 4x^3 – 24x^2 + 44x – 24 and f”(x) = 12x^2 – 48x + 44  ……….(2)
 
At an extremum of f(x), we have f’(x) = 0
 
Or 4x^3 – 24x^2 + 44x – 24 = 0
 
Or x^2(x – 1) – 5x(x – 1) + 6(x – 1) = 0
 
Or (x – 1)(x^2 – 5x + 6) = 0
 
Or (x – 1)(x – 2)(x – 3) = 0
 
So, x = 1, 2, 3
 
Now, f”(x) = 12x^2 – 48x + 44
 
f”(1) = 8 > 0
 
f”(2) = -4 < 0
 
f”(3) = 8 < 0
 
So, f(x) has maximum at x = 2.