1549901369-Elements_of_Real_Analysis__Denlinger_

6.2 Rules for Differentiation 311 By the sequential criterion for derivatives, (f-^1 )' (J(x 0 )) = -( 1 )' • f' Xo Example 6.2. ...

312 Chapter 6 11 Differentiable Functions Corollary 6.2. 8 l::/r E Q, the power function f(x) = xr is differentiable every- wher ...

6.2 Rules for Differentiation 313 Corollary 6.2.10 Suppose a> 0, and a -=j:. l. Then d d 1 (a)VxEIR,-d ax=axlna; (b)Vx>O, ...

314 Chapter 6 11 Differentiable Functions Prove that if f is differentiable at xo and f (xo) =f. 0, then _!}__ [-1 ] -f'(x) dx ...

6.2 Rules for Differentiation 315 Prove the general power rule for real number exponents: If f is differen- tiable then v c E J ...

316 Chapter 6 • Differentiable Functions (c) if f is continuous on [a, b) and lim f(x) = f(a), then Jis continuous x---+b- on R ...

6.3 Local Extrema and Monotone Functions 317 Theorem 6.3.2 Suppose f'(xo) > 0 at an interior point xo of V(f). Then :Jo>03 ...

318 Chapter 6 • Differentiable Functions Proof. Exercise 1. • Theorem 6.3.4 (Local Extreme Value Theorem) If a function f has a ...

6.3 Local Extrema and Monotone Functions 319 Note: In the next section, after we have proved the mean value theorem, we shall pr ...

320 Chapter 6 • Differentiable Functions Thus, by (a) and (b), g must have its maximum value for [a, b] at some point c E (a, b) ...

6.4 Mean-Value Type Theorems 321 Prove that there is no differentiable function f, defined on any open interval I containing 0 ...

322 Chapter 6 • Differentiable Functions y x Figure 6.6 Proof. Suppose f is differentiable on (a, b) and continuous on [a, b], w ...

6.4 Mean-Value Type Theorems 323 Theorem 6.4.3 (Mean Value Theorem, "MVT") Suppose f: [a, b] -t JR is differentiable on (a,b), a ...

324 Chapter 6 • Differentiable Functions Thus, h'(c) = 0::::} f'(c) - f(b) - f(a) = 0, and hence, b-a f'(c) = f(b~ =~(a). • APPL ...

6.4 Mean-Value Type Theorems 325 The result we have just proved should be quite familiar to you. Where have you seen it before? ...

326 Chapter 6 • Differentiable Functions Solution. Let x, y E R If x = y, the desired inequ ality is true, since both sides are ...

6.4 Mean-Value Type Theorems 327 Prove that the equation 3x^4 - 8x^3 + 7x^2 - 45 = 0 has exactly two real roots. [You must pro ...

328 Chapter 6 • Differentiable Functions By Exercise 21 , if f is differentiable and has a continuous derivative in some neighb ...

6.5 Taylor's Theorem 329 (a) Prove that if f' exists and is bounded on an interval I (possibly infinite) then f satisfies a Lip ...

330 Chapter 6 • Differentiable Functions That is , n J(k)(a) k Tn(x) = 2:-k,-(x - a) , k=O (4) where j<^0 J = f and J(k) deno ...