Derivatives 2

3 1) f'(x) = 5x4 + 10 > 0 for all x

Also lim (x → -∞) f(x) = - ∞ and lim (x → +∞) f(x) = + ∞

Hence f(x) crosses the x-axis exactly once.

f(x) = 0 has exactly one real root.

3 2) f'(x) = 3 - (π/2) sin (πx/2) > 0 since 3 > π/2

Also lim (x → -∞) f(x) = - ∞ and lim (x → +∞) f(x) = + ∞

Hence f(x) crosses the x-axis exactly once.

f(x) = 0 has exactly one real root.

4) Using mean value theorem, there exists -5 < c < 5 such that:

[f(5) - f(-5)]/10 = f'(c) <= 1

[f(5) + 10]/10 <= 1

f(5) + 10 <= 10

f(5) <= 0

Hence f(5) has a greatest possible value of 0.