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  3. 1. (8pt) consider the function f(x) = (x + 3)(x +

1. (8pt) Consider the function f(x) = (x + 3)(x + 5)^2 √ (7 −...

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1. (8pt) Consider the function f(x) = (x + 3)(x + 5)^2 √ (7 − x)whose first and second derivatives are f'(x) = ((x + 5)(139 + 12x −7x^2))/2 √ (7 − x) , f''(x) = (35x^3 − 225x^2 − 843x + 3481)/ 4(7 −x)^3/2 .

Note: 35x 3 − 225x 2 − 843x + 3481 has three roots: x1 ≈ 7.8835,x2 ≈ −4.3531 and x3 ≈ 2.8982.

(a) (1pt) What is the domain of f(x)? (b) (1pt) What are thex-intercepts and y-intercepts of f(x)? (c) (1pt) What are thecritical numbers of f(x)? (d) (1pt) On what open interval(s) isf(x) increasing, and on what open interval(s) f(x) is decreasing?(e) (1pt) What are the x-values of potential points of inflectionof f(x)? (f) (1pt) On what open interval(s) is f(x) concave-upward,and on what open interval(s) f(x) is concave-downward? (g) (2pt)Sketch y = f(x). Clearly label the intercepts, extrema, and pointsof inflection.

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