Question

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] ?

Answer 1

Answer:

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