### Blog Assignment 1

A differentiable function f defined on -7 < 0< 7 has f(0)=0 and f'(x) = 2x sin x- e^(-x^2) +1

a) describe the symmetry of f.

b) On what intervals is f decreasing?

c) For what values of x does f have a relative maximum? Justify your answer.

d) How many points of inflection does f have? Justify your answer.

a) f is an even function.

d) f"(x)=2x cos x+ 2 sinx + 2x e^(-x^2)

2x cosx+2 sin x +2x e^(-x^2) = 0

x= -4.9136, -2.0405, 0, 2.0405, -4,9136

f has inflection points when f"(x)= o, so, f has 5 inflection points

a) describe the symmetry of f.

b) On what intervals is f decreasing?

c) For what values of x does f have a relative maximum? Justify your answer.

d) How many points of inflection does f have? Justify your answer.

a) f is an even function.

b) f'(x) = 2x sinx - e^(-x^2) + 1

2x sinx- e^(-x^2) +1 =0

x=-6.2024, -3.294, 0, 3.294, 6.2024

+ - + + - +

--------1----------1----------1-----------1-----------1-----------

-6.2024 , -3.294 , 0 , 3.294 , 6.2024

If f'(x) less than zero,f(x) is decreasing, therefore, f(x) is decreasing at (-6.2024, -3.294) and (3.294, 6.0224).'

c) If f'(x) larger than zero, f(x) is increasing; If f'(x) less than zero, f is decreasing.

from increasing to decreasing, f reaches a maximum. f has relative maximum at x= -6.2024, and x= 6.2024

d) f"(x)=2x cos x+ 2 sinx + 2x e^(-x^2)

2x cosx+2 sin x +2x e^(-x^2) = 0

x= -4.9136, -2.0405, 0, 2.0405, -4,9136

f has inflection points when f"(x)= o, so, f has 5 inflection points

## 1 Comments:

At 8:50 AM CDT, Mr. Kuropatwa said…

(a) f' is even and f is odd. Can you figure out why? (Do you know any points on f?)

(b) The second interval of decrease should be (3.294, 6.202). Can you figure out why or was it a typo?

(c) The positive relative maximum is at x = 3.294 not x = 6.202. Can you figure out why?

(d) Correct!

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