Applications of Differentiation

Question

Consider the function f(x)=x^(9) e^(9x) , -2<= x <=1.

The absolute maximum value is =

and this occurs at x equals =

The absolute minimum value is =

 and this occurs at x equals =

Answer 1

f(x) = x⁹e^9x, where -2 ≤ x ≤ 1

f'(x) = 9x⁸e^9x + x⁹(9e^9x) = 9x⁸e^9x(1 + x)

f'(x) = 0 when x = 0 or -1

The absolute extrema occur at a critical point or at an endpoint of the interval.

f(-1) = -e^-9

f(0) = 0

f(-2) = (-2)⁹e^-18 (absolute minimum)

f(1) = e⁹ (absolute max)