Question

Find the absolute extrema if they​ exist, as well as all values of x where they​ occur, for the function:

f(x) = x + e^3x ; on the domain [-3 , 2]

Answer 1

f(x) = x + e^(3x) in interval [-3, 2]

f'(x) = 1 + 3*e^(3x)

Extreme points will be at

f'(x) = 0 and at boundary of given interval, So

1 + 3*e^(3x) = 0

e^(3x) = - 1/3

Since, e^x > 0

So, e^(3x) 0 at any value of x.

then, critical point = None

Now calculate the max and min value of function at its boundry points.

f(x) = x + e^(3x)

at x = -3

f(-3) = -3 + e^(-3*3) = -3

at x = 2

f(2) = 2 + e^(3*2) = 405.43

So,

absolute minimum will be at x = -3, which is f(-3) = -3 + e^-9 , (-3, -3+e^-9)

absolute maximum will be at x = 2, which is f(2) = 405.43 , (2, 405.43)