Final answer:
To find the value of c for which f'(π/4) = 9, we need to find the derivative of f(x) = cx ln(cos(x)) and evaluate it at π/4. The value of c is approximately -24.932.
Step-by-step explanation:
To find the value of c for which f'(π/4) = 9, we need to find the derivative of f(x) = cx ln(cos(x)) and evaluate it at π/4.
Let's start by finding the derivative:
f'(x) = c ln(cos(x)) + cx * (1/cos(x)) * (-sin(x))
Now, substitute x = π/4 into f'(x) and set it equal to 9:
c ln(cos(π/4)) + c(π/4) * (1/cos(π/4)) * (-sin(π/4)) = 9
Simplifying:
c(ln(√2/2) - (√2/4)) = 9
Using the given information that ln(√2/2) = 0.346, we can solve for c:
c(0.346 - 0.707) = 9
After simplifying, we find that c = 9/(-0.361). Therefore, the value of c for which f'(π/4) = 9 is approximately -24.932.