Question

let -pi/2 < x < pi/2

f(x)=e^x*sin(x)

(a) Find the gradient of y=f(x) when x=pi/4

(b) Find the coordinates of the point where the gradient is zero

I know part (a) but im struggling in part (b). Could someone please help explain it to me. Thx

Answer 1

Is it not that gradient mean derivative??

Answer 2

and for part b, just let it =0 solve for x then plug back to y to solve for y.

Answer 3

\[f'(x) = e^x(sinx + cosx)=0\]

Answer 4

\(e^x\neq 0~~\forall x\) --> sinx + cos x =0 only. I think you can step up from here, right?

Answer 5

no thats exactly where i stopped :)

Answer 6

@Loser66

Answer 7

what do you mean? that is part b

Answer 8

yes

Answer 9

you take gradient of f(x) means derivative of f(x) , right?

Answer 10

its asking me to find the coordinates

Answer 11

yes, it is.

Answer 12

the answer in the text book is (-pi/4, -sqrt2/2*e^-pi/4)

Answer 13

now solve for sinx + cos x=0 , you have x = -pi/4 for example

Answer 14

now, plug x = -pi/4 into f(x) you have f(x) = e^x sin x

Answer 15

where did x=-pi/4 come from?

Answer 16

from sin x = -cosx

Answer 17

could you please explain it to me step by step

Answer 18

ok, restart

Answer 19

what is the gradient of f(x)? in general??

Answer 20

f'(x)

Answer 21

=?

Answer 22

f'(x)=e^x(sinx+cosx)

Answer 23

yes, now, they ask " where it is =0 " means f'(x) =0 at x = somewhere, right?
so, just let it =0 and solve for x

Answer 24

e^x(sinx+cosx)=0

Answer 25

and e^x \(\neq 0\) for all x, right?

Answer 26

right

Answer 27

so, to get it is =0 , you have only 1 option, that is sinx + cos x =0, right?

Answer 28

sinx= -cos x

Answer 29

in the interval \([-\pi/2,\pi/2]\) , which value gives you sinx =-cos x??

Answer 30

7pi/4

Answer 31

is it not that 7pi/4 = -pi/4??

Answer 32

correct

Answer 33

or you can see it from
You have to take this value because 7pi/4 is out of the interval

Answer 34

got me so far?

Answer 35

yes

Answer 36

now, at x = -pi/4 , f' (x) =0 ( remember, just f' not f)
so, if x = -pi/4, y =? ( y is f(x)

Answer 37

do i subtract 2pi from it?

Answer 38

no need to do, just take the logic of 7pi/4 = -pi/4

Answer 39

ok

Answer 40

ok, answer me, x = -pi/4, f(x) =?

Answer 41

i plug it in

Answer 42

yup

Answer 43

i get -sqrt2/2*e^-pi/4

Answer 44

thats the exact answer in the textbook

Answer 45

thank you so much for your help and time

Answer 46

You got the concept, right?

Answer 47

yes :)

Answer 48

good