Question

Find the local extrema for the following functions using second derivative test:

f(x) = – 3x⁵ + 5x³

Answer 1

f(x) = – 3x⁵ + 5x³

f'(x) = 0, f”(x) = – ve at x = a

⇒ x = a is a maximum point

f'(x) = 0, f”(x) = + ve at x = 6

⇒ x = b is a minimum point

f(x) = – 3x⁵ + 5x³

f’(x) = – 15x⁴ + 15x²

f”(x) = – 60x³ + 30x

f'(x) = 0

⇒ – 15x² (x² – 1) = 0

⇒ x = 0, +1, – 1

At x = 0, f”(x) = 0

At x = 1, f”(x) = – 60 + 30 = – ve

At x = – 1, f”(x) = 60 – 30 = + ve

So at x = 1, f'(x) = 0 and f”(x) = – ve

⇒ x = 1 is a local maximum point.

And f(1) = 2

So the local maximum is (1, 2)

At x = – 1, f'(x) = 0 and f”(x) = + ve

⇒ x = – 1 is a local maximum point and f(– 1) = – 2.

So the local minimum point is (– 1, – 2)

∴ a local minimum is – 2 and the local maximum is 2.