No, not really.

You just need to continue.

am i supposed to use the zero product rule for
-sinx-cosx=0?
-sinx=0 or -cosx=0
but with the negative sign infront of sinx and cosx, i dont really know how to proceed...

-sin x - cos x = 0
sin x = -cos x
sin x / cos x = -1
tan x = -1

This should be familiar. It's a special angle. tan(π/4) = 1 but that's for first and third quadrant.

We need to look at the second and fourth quadrant for tan x = -1

Since they want positive values of x,

x = 180° - 45°, 360° - 45°, 180° - 45° + 360°, 360° - 45° + 360°,...

x = 135°, 315°, 495°, 675°

Or

x = π - π/4, 2π - π/4, π - π/4 + 2π, 2π - π/4 + 2π,...

x = 3π/4, 7π/4, 11π/4, 15π/4,...

Smallest positive value of x for which there is a stationary point

= 135° or 3π/4 radians

No, you cannot let either be 0 since it's not a product but it's a subtraction

i.e it's -sin x - cos x = 0

NOT

-sin x (- cos x) = 0
sin x cos x = 0

Hence part :

Your dy/dx expression can be simplified into :

-(sin x + cos x) / e^x

We know that e^x is positive for all real x so we just need to check the sign of -(sin x + cos x) to the left and right of x = 3π/4 rad (or 135°)
(First derivative test)


We can just look at the isosceles right-angled triangle obtained when x = 3π/4 rad (it is the second quadrant)

Here, tan x = -1 so the opposite and adjacent side have the same absolute length, just that the coordinates for the adjacent are negative.

sin x = - cos x here since :
sin x = opp/hyp = -adj/hyp = -cos x


①
Now just increasing the angle by a little bit causes the adjacent to lengthen by a bit, while the opposite gets a little bit shorter (while keeping the hypotenuse fixed)

sin x decreases by a little (becomes less positive , cos x becomes more negative (since the adjacent is more negative now)


so sin x + cos x < 0
-(sin x + cos x) > 0
dy/dx > 0

② the opposite occurs when we decrease the angle a bit. Adjacent gets shorter, opposite side gets longer (while keeping the hypotenuse fixed)

sin x + cos x > 0
-(sin x + cos x) < 0
dy/dx < 0


So we can see this is a minimum point.

I used to have a small proportion of students who erroneously thought that -a - b means -a (-b) or -a times -b (an over-usage of brackets), but lately I have not seen this error being made.

Students just need to remember that the minus sign(or any sign that comes before it) is an operater, done to the number or letter.

-5 - 4 = -9
-5 + 3 = -3

The -5 in this case should be thought of as (0 - 5) if needed.

If multiplication is needed then brackets will be placed

(-5) × (-4) or (-5)(-4)

-5 × -4 (not recommended as two operators should not be next to each other without brackets)