Let f be the function given by f (x) = 5cos2 (x2) + ln (x+1) - 3. The derivative of f is given by f′ (x) = -5cos (x2) sin (x2) + 1x+1. What value of c satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4]?

The value of c that satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4] is approximately 0.1793.

Step-by-step explanation:

Given f (x) = 5cos²x² + ln (x + 1) - 3

The Mean Value Theorem applied to f on an interval [a, b] is given as

f' (x) = c = [f (b) - f (a) ] / (b - a)

= [f (4) - f (1) ] / (4 - 1)

c = [f (4) - f (1) ]/3

f (4) = 5 (cos4²) ² + ln (5) - 3

= 3.2296

f (1) = 5 (cos1²) ² + ln (2) - 3

= 2.6916.

c = (3.2296 - 2.6916) / 3

= 0.538/3

≈ 0.1793