I understand how to do mean value theorum but I'm not sure how to apply it with ln(x).
f(x)=ln(x), [1,8]
How can I find a c that satisfies the conclusion of the Mean Value theorem by using ln(x)?
I know its f(b)−f(a)b−a, then take derivative and fill in the slope.
But how do I solve this with ln? I only did this with quadratic.
Answer
The mean value theorem states that if f(x) is continuous on an interval [a,b] and differentiable on (a,b), then there exists a c∈(a,b) such that f′(c)=f(b)−f(a)b−a
In your case, the function f(x)=ln(x) is continuous on an interval [1,8] and differentiable on (1,8). The derivative of ln(x) is 1x in the interval (1,8). Hence, by mean value theorem, ∃c∈(1,8) such that f′(c)=1c=f(8)−f(1)8−1=ln(8)−ln(1)8−1=3ln(2)−07=3ln(2)7
Hence, the desired point c is 73ln(2).
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