Question

f(x)=eˣ/x²+15​ f′(3)= Remairing time: 43 : 53 Iminibes Problem 12. (t) poinc) thelect all that apply) A. A. Abooldte maximum a. Rielation minimum C. Absclute minimum D. Relative maximum E. None of the above A. Piblative mavirum B. Absolute miniencim 6. Felative minimum D. Abscile maximurn

Answer 1

The function
`\( f(x) = (e^x)/(x^2 + 15) \)` has a relative minimum at x = 3.The correct option is D. Relative maximum.

Step-by-step explanation:

In order to determine the nature of the critical point x = 3 for the given function f(x) , we need to find its derivative f'(x) and evaluate it at x = 3 ).

The derivative of ( f(x) with respect to x is calculated using the quotient rule:

`\[ f'(x) = ((e^x \cdot (x^2 + 15)) - (2x \cdot e^x))/((x^2 + 15)^2) \]`

After obtaining the derivative, we substitute x = 3 to find f'(3) . This value represents the slope of the tangent line to the graph of f(x)at x = 3 .

Next, we analyze the sign of f'(3) . If f'(3)is negative before x = 3 and positive after x = 3 then f(x) has a relative minimum at x = 3 .

In conclusion, after performing the calculations, we find that f'(3) is indeed negative before x = 3 and positive after x = 3 , indicating a change from decreasing to increasing. Therefore, f(x) has a relative minimum at x = 3. The options corresponding to a relative minimum, D. Relative maximum and E. None of the above , are selected.

The correct option is D. Relative maximum.