Question

Let f(x) = x⁴ – 4x³ + 10. If m is the number of point of inflection and n is the number of the points of extremum, find the value of (m + n + 1).

Answer 1

We have f(x) = x⁴ – 4x³ + 10

⇒ f'(x) = 4x³ – 12x²

⇒  f''(x) = 12x² – 24x

Now, f''(x) = 0 gives 12x² – 24x = 0

⇒ 12x (x – 2) = 0

⇒ x = 0, 2

Thus, the number of point of inflection is 2

So, m = 2

Also, f'(x) = 0 gives 4x³ – 12x² = 0

⇒ 4x²(x – 3) = 0

⇒ x = 0, 3

Thus, the number of point of extrema is 1

at x = 3, since x = 0 is the point of inflection

So, n = 1

Hence, the value of (m + n + 1)

= 2 + 1 + 1

= 4