Suppose a certain real-valued function f is continuous on the interval [35, 38] and differentiable on (35, 38). Moreover, suppose we also know f'(x) -190 in the essay box below.Suppose a certain real-valued function f is continuous on the interval [35, 38] and differentiable on (35, 38) Moreover, suppose we also know f' (x) -190 in the essay box below.Suppose a certain realvalued function f is continuous on the interval [35, 38] and differentiable on [351 38]. Moreover, suppose we also know f! [3} E 95, for all a: E [351 38], and 37) : 2_ [a] We wish to nd an explicit upper bound for 33]. Complete the following proof: Proof. First, since f is continuous on the interval Click for List - and differentiable on Click for List ' then. by CIicI-cforList - . Click for List - I1 = [31338] 38 37 ' Rearranging this and applying our assumptions on f, we conclude that f[38] = f(3?) + f'rc} g 2 + 95 = 93. This completes the proof. {b} Using a similar argument, prove that f(35) 3 191] in the essayr box below. Suppose a certain realvalued function f is continuous on the interval [35, 38] and differentiable on (35, 38]. Moreover, suppose we also know 3" [3} E 95, for all 2: E [35,38], and f(3?) = 2 (a) We wish to nd an explicit upper bound for f{38]. Complete the following proof: Proof. First, since f is continuous on the interval Click for List - and differentiable on Click for List then, by Click forList - . Click for List - the intermediate value theorem f(33) _ f[3'f) I = c _ the mean value theorem 33 37 f ( J 'ons on f, we conclude that 2 g?) +f'{c} g 2 +96 2 93. Rearran Rolle's theorem the fundamental theorem of calculus This co l'Hpri'taI's rule Suppose a certain realvalued function f is continuous on the interval [35, 38] and differentiable on [351 38]. Moreover, suppose we also know 1" (I) :1 96, forall a: 6 [35,33], and f(37) = 2. (a) We wish to nd an explicit upper bound for 38]. Complete the following proof: Proof. First, since f is continuous on the interval Click for List - and differentiable on elicit for List - then, by Click for List - CIicI-c for List there exists some .2. E II such that for all c E 1'], we have C 11- 'I- I I" 3 Rearranging this and applying our assumpti- f[38] = 313?) + rst g 2 + 96 = 93. This completes the proof. {b} Using a similar argument, prove that 35) 2 19l]I in the essayr box below