BU111 Lecture 7: Lab Report 7 Solutions.pdf
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Ma129 lab report 7 - shapes of curves and extrema continued. [4 marks] (a) suppose that f (x) = g(x)h(x): show that f00(x) = g00(x)h(x) + 2g0(x)h0(x) + g(x)h00(x): f0(x) = f00(x) = d dx d dx. = [g00(x)h(x) + g0(x)h0(x)] + [g0(x)h0(x) + g(x)h00(x)] = g00(x)h(x) + 2g0(x)h0(x) + g(x)h00(x) (b) use the result from part (a) to nd f00(x) when f (x) = xex: f00(x) = 0 (cid:1) ex + 2(1)ex + xex = 2ex + xex. [7 marks] consider the function f (x) = 64 ln x (cid:0) x4 for x 2 (0;1): (a) use the second derivative test to nd any relative extrema of f: f0(x) = Critical values of f on (0;1) : f0(x) = 0 ) 64 (cid:0) 4x4 = 0. Now: f00(x) = (cid:0)16x3(x) (cid:0) (64 (cid:0) 4x4)(1) ) x = 2 (as x = (cid:0)2 =2 (0;1)) = (cid:0)12x4 (cid:0) 64 x2 x2 f00(2) = (cid:0)12(16) (cid:0) 64.
