Question

Find the points of local maxima, local minima and the points of inflection of the function f(x) = x⁵ – 5x⁴ + 5x³ – 1. Also find the corresponding local maximum and local minimum values.

Answer 1

We have f(x) = x⁵ – 5x⁴ + 5x³ – 1

⇒ f '(x) = 5x⁴ – 20x³ + 15x²

For local maxima and local minima, f '(x) = 0

⇒ 5x⁴ – 20x³ + 15x² = 0

⇒ 5x²(x² – 4x + 3) = 0

⇒ 5x²(x² – 3x – x + 3) = 0

⇒ x²(x – 3)(x – 1) = 0

∴ x = 0, x = 1 and x = 3

Now f '(x) = 20x³ – 60x² + 30x

⇒ `"f''"(x)_("at" x = 0)` = 20(0)³ – 60(0)² + 30(0) = 0

Which is neither maxima nor minima.

∴ f (x) has the point of inflection at x = 0

`"f''"(x)_("at" x = 1)` = 20(1)³ – 60(1)² + 30(1)

= 20 – 60 + 30

= –10 < 0 Maxima

`"f''"(x)_("at" x = 2)` = 20(3)³ – 60(3)² + 30(3)

= 540 – 540 + 90

= 90 > 0 Minima

The maximum value of the function at x = 1

f (x) = (1)⁵ – 5(1)⁴ + 5(1)³ – 1

= 1 – 5 + 5 – 1

= 0

The minimum value at x = 3 is

f (x) = (3)⁵ – 5(3)⁴ + 5(3)³ – 1

= 243 – 405 + 135 – 1

= 378 – 406

= – 28

Hence, the function has its maxima at x = 1 and the maximum value = 0 and it has minimum value at x = 3 and its minimum value is – 28.