To find the critical points of f(x) = x^16 * e^-x, we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
First, let's find the derivative of f(x):
f'(x) = 16x^15 * e^-x - x^16 * e^-x
Now, let's set the derivative equal to zero and solve for x:
16x^15 * e^-x - x^16 * e^-x = 0
e^-x(16x^15 - x^16) = 0
Since e^-x is never zero for any value of x, the only way for the above expression to be zero is if 16x^15 - x^16 = 0.
Next, let's factor out x^15:
x^15(16 - x) = 0
Setting each factor equal to zero:
x^15 = 0 --> x = 0
16 - x = 0 --> x = 16
So, the critical points of f(x) = x^16 * e^-x are x = 0 and x = 16.
a. Find the critical points of f(x)=x^16
e^-x
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Determine the intervals on which f is concave up or concave down. Identify any inflection points.
a. Identify all the critical points. Select the correct choice below and fill in any answer boxes within your choice.
1 answer