For the function f(x)=x^5-5x^4+5x^3-1 , which of the following are tr…

For the function $ f(x)=x^{5}-5x^{4}+5x^{3}-1 $, which of the following are true?

(A) $ f(x) $ has local minimum value equal to -28.

(B) $ f(x) $ has point of inflection at x = 1.

(D) $ f(x) $ has the local maximum value equal to 2.

Choose the correct answer from the options given below:

Answer & explanation

Correct answer: option 3

The correct answer is Option (3) → (A) and (C) only

$ \displaystyle f'(x)=5x^4-20x^3+15x^2 $

$ \displaystyle =5x^2(x^2-4x+3) $

$ \displaystyle =5x^2(x-1)(x-3) $

Critical points: x = 0, 1, 3. Since $ 5x^2\ge 0 $, the sign of f'(x) is the sign of (x − 1)(x − 3):

    $ \displaystyle x\lt 0:\ + $
    $ \displaystyle 0\lt x\lt 1:\ + $
    $ \displaystyle 1\lt x\lt 3:\ - $
    $ \displaystyle x\gt 3:\ + $

So f' does not change sign at x = 0 (no extremum), changes from + to − at x = 1 (local maximum) and from − to + at x = 3 (local minimum).

$ \displaystyle f(1)=1-5+5-1=0 $
$ \displaystyle f(3)=243-405+135-1=-28 $

$ \displaystyle f''(x)=20x^3-60x^2+30x $
$ \displaystyle f''(1)=20-60+30=-10 $

So x = 1 is a point of local maximum with local maximum value f(1) = 0, x = 3 is a point of local minimum with local minimum value f(3) = −28, and f''(1) = −10 ≠ 0.

Checking the statements

(A) $ f(x) $ has local minimum value equal to -28. — Correct. f(3) = −28.

(B) $ f(x) $ has point of inflection at x = 1. — Incorrect. x = 1 is a local maximum, and f''(1) = −10 ≠ 0.

(C) $ f(x) $ has point of local maxima at x = 1. — Correct. f' changes from + to − at x = 1.

(D) $ f(x) $ has the local maximum value equal to 2. — Incorrect. The local maximum value is f(1) = 0.

NCERT Exemplar:
Application of Derivatives

$NCERT extract$