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.
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.