Mean Value Theorem a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a. b]. b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. c. Make a sketch of the function and the line that passes through (a.f(a)) and (b,f(b)). Mark the points P (if they exist) at which the slope of the function equals the slope of the secant line. Then sketch the tangent line at P. f(x) =x+2 on [-1,2]
Solution 22E Step 1 In this problem we have to check whether the mean value theorem is applicable for x f(x) = x+2on [-1,2]. And if it is applicable we have to find a point that guarantees the mean value theorem and then we have to draw a sketch of the function with given conditions. Mean Value theorem: If f is defined and continuous on the closed interval [a,b]and differentiable on the open interval (a,b)then there is at least one point cin (a,b)that is a < c < bsuch that f (c) = f(b) f(.) ba a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a,b]. x Given f(x) = x+2 (x+2)1 x(1) f (x) = (x+2)2 = x+2x (x+2) 2 = (x+2)2 Since f (x)is finite for all values of x in the interval (-1,2), f(x)is differentiable in (-1,2) … (1) We know that “Every differentiable function is continuous”. f(x)is continuous in [-1,2] …. (2) From (1) and (2) we get f(x)satisfies all the requirements of mean value theorem. x Hence mean value theorem is applicable for f(x) = x+2 on [-1,2].