Question

Use first derivative analysis (no calculators) to graph each function. (By first derivative analysis we mean the following as demonstrated in class: find critical values indicate whether the first derivative is 0 (producing a horizontal tangent) or undefined (producing sharp corner or vertical tangent) at each critical value o o o show tables of intervals where f increases or decreases and thus whether critical values correspond to a local maximum, local minimum, or neither). x) (4-x2)

Answer 1

(a) Differentiate,f(x)=(4-x2)" with respet tox (x)(4-(-2104-2) 厂(x) 2 0 for x 0,厂(x) 0for x > 0 and f'(±2)=0 Interval of increase: (-00,0 Interval of decrease (0,o) 厂(x) doesn't change sign at x =-22 where asf"(0) > 0 andf"(r) < 0 so x = 0 is point of local maxima. The graph of the function f(x) is shown below (b) Differentiate,f(x) . xs_ax" + 16x with respet to x 厂(x)-0 → x =-22 and (x) > 0 elsewhere The crtical points are x =-22 and 2-0 The sign off' (x) desn't change at x =-22, so there is no local maximiminima. The function is always increasing as (x)2 0 Since (x)2 0 andf(0) = 0, so the graph of f(x) les below the axis for x < 0 and above the x-axis for x>0 The graph of the functionf(x) is shown below:

10 500 -3 -2 -500 -1000

plot (8x3) + 16 x x =-3 to 3 Plot: 20 10 -3 -10 -20