The Mean Value Theorem and its Applications Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math229 1. Extreme Value Theorem Assume f(x) is a continuous function defined on a closed interval [a,b]. Then there are numbers c and d in the interval [a,b] such that f(c) = the absolute minimum and f(d) = the absolute maximum. 2. Vanishing Derivative Theorem Assume f(x) is a continuous function defined on an open interval (a,b). Assume that f(x) has a local maximum or minimum at a point c inside (a,b). If f (c) exists, then f (c) = 0. 3. Max-Min Theorem for Closed Intervals Assume f(x) is a continuous function defined on a closed interval [a,b]. Assume that f(x) has a local maximum or minimum at a point c in [a,b]. Then (i) f (c) = 0 (ii) f (c) is undefined or (iii) c is an endpoint a or b. 1
2 4. Rolle s Theorem Assume f(x) is a continuous function defined on a closed interval [a,b]. Assume further that f(x) is differentiable on the open interval (a,b). If f(a) = f(b), then for some point c between a and b, f (c) = 0. Less formally, if a smooth function has two y values which are the same, say f(a) = f(b), then somewhere between a and b its graph turns around, that is, there is point at which the derivative is 0. 5. Proof of Rolle s Theorem By the Extreme Value Theorem, the function has an absolute maximum and an absolute minimum. If either of these occur at a point c inside the closed interval, then by the Vanishing Derivative Theorem f (c) = 0 and we are done. (Note that our hypothesis guarantees that f (c) is defined.) One case left to consider: neither the absolute maximum nor the absolute minimum lies inside (a, b). By the Max Min Theorem, these extreme points must both occur at the two endpoints a and b. So one endpoint is the absolute maximum of the function and the other is the absolute minimum. But by hypothesis f(a) = f(b). If the absolute max and the absolute min have identical values, then the function f(x) must be a constant function on the interval [a,b]. The derivative of a constant function is always zero, so any c inside (a,b) must have a 0 derivative.
3 6. The Mean Value Theorem The idea is to generalize Rolle s Theorem by considering the case when f(a) does not necessarily equal f(b). Hypothesis: Assume f(x) is a continuous function defined on a closed interval [a,b]. Assume further that f(x) is differentiable on the open interval (a,b). Then for some point c between a and b, f (c) = f(b) f(a). 7. MVT versus Rolle s Theorem Note that Rolle s Theorem is a special case of the Mean Value Theorem, since when f(a) = f(b), we get f(b) f(a) = f(a) f(a) = 0. It turns out that we can use Rolle s Theorem to prove the Mean Value Theorem. The idea involves some algebraic trickery. Define 8. Proof of the MVT g(x) = f(x) f(b) f(a) (x a). Plugging in x = a and x = b gives g(a) = f(a) f(b) f(a) (a a) = f(a) while g(b) = f(b) f(b) f(a) () = f(a). So g(a) = g(b). By Rolle s Theorem g (c) = 0 for some point c between a and b.
4 g(x) = f(x) f(b) f(a) (x a) and 9. Proof Continued g (c) = 0 for some c between a and b. By the rules of derivatives, g (c) = f (c) f(b) f(a) = f (c) = f(b) f(a) g (x) = f (x) f(b) f(a) = 0 1 10. Example Given f(x) = 3 6. Find all c in the interval (2,6) such that x f (c) = f(6) f(2) 6 2 First, f(6) = 3 1 = 2 and f(2) = 3 3 = 0. So f(6) f(2) 6 2 = 2 0 6 2 = 1 2. f(x) = 3 6x 1 So f (x) = 6( 1)x 2 = 6 x 2. f (c) = f(6) f(2) 6 2 becomes 6 c 2 = 1 2 or c 2 = 12 11. Example Continued
5 Answer c = 12 Note that 12 lies in the interval (2,6). Why? 12. Geometric Interpretation By Calculus, f (c) is the slope of the tangent line at the point (c,f(c)). The fraction f(b) f(a) represents the slope of the line segment joining points (a,f(a)) and (b,f(b)). Thus the Mean Value Theorem says that under the right conditions, the tangent line to the curve will be parallel to this line segment. Draw some pictures! 13. Application to velocity If you think of s = f(t) as a distance function at time t, then the fraction f(b) f(a) represents your average velocity in travelling from time t = a to time t = b. The Mean Value Theorem asserts that at some point on your journey, the instantaneous velocity (registered on your speedometer) is precisely your average velocity. If you average 70 mph on a trip, then at some time your speedometer must read exactly 70 mph. Verify the following statement: 14. Theoretical Value of MVT [Zero Derivative Theorem] Assume f(x) is a continuous function defined on a closed interval [a,b]. Assume further that f(x) is differentiable on the open interval (a,b). If f (x) = 0 for every x in (a,b), then f is constant throughout [a,b]. Note that this statement does not say that the derivative of a constant function is 0.
6 Rather it says that under the appropriate conditions, if the derivative of f is 0, then f must be a constant function. 15. Proof of Zero Derivative Thm Take any two points x 1 < x 2 in the interval [a,b]. By the MVT, for some point c between x 1 and x 2, f(x 2 ) f(x 1 ) x 2 x 1 = f (c) = 0 How do we know f (c) = 0? We conclude immediately that f(x 2 ) = f(x 1 ). Why? A function whose y-values on any two x-values are the same is a constant function.