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# Bah, more proofs. watch

1. 1. Given that y=2e^-4x - e^3x, prove that
a) y<1 for x>0
b) d^2y/dx^2 + 2dy/dx - 8y = -7e^3x

2. Prove that the curve with equation y=4x^5+lambdax^3 has two turning points for lambda <0 and none for lambda >>0. Given that lambda = -5/3, find the coordinates of the turning points and distinguish between them.
2. Given that y=2e^-4x - e^3x, prove that
a) y<1 for x>0and distinguish between them.

you need to show that if x is positive,
2 < e^7x + e^4x

try induction.
although I'm not sure whether this works when x can take real values, as it's only for integers...
3. (1a)
If x > 0 then e^(-4x) < 1 and e^(3x) > 1. This is because e^(anything positive) > 1 and e^(anything negative) < 1.

So y equals (something less than 2) minus (something more than 1). So y < 1.

(1b)
dy/dx = -8e^(-4x) - 3e^(3x)
d^2y/dx^2 = 32e^(-4x) - 9e^(3x)

So the left-hand side
= 32e^(-4x) - 9e^(3x) + 2(-8e^(-4x) - 3e^(3x)) - 8(2e^(-4x) - e^(3x))
= -7e^(3x)
= the right-hand side.
4. (Original post by mik1a)
Given that y=2e^-4x - e^3x, prove that
a) y<1 for x>0and distinguish between them.

you need to show that if x is positive,
2 < e^7x + e^4x

try induction.
although I'm not sure whether this works when x can take real values, as it's only for integers...
use calculus to show either 1) y has a maximum value which is less than 1 or 2) dy/dx < 0 for all x, In this example use method 2 and that y(0) = 1 so y has a maximum value of 1 over the range x>0
5. (2)
Remember that having dy/dx = 0 doesn't necessarily mean that we have a turning point.

dy/dx
= 20x^4 + 3 lambda x^2
= x^2 (20x^2 + 3 lambda).

If lambda < 0 then dy/dx = 0 for three values of x:

(i) x = 0,
(ii) x = sqrt(3 lambda/20),
(iii) x = -sqrt(3 lambda/20).

Option (i) is not a turning point because dy/dx is negative on both sides of the value. Options (ii) and (iii) are turning points because dy/dx is negative on one side of the value and positive on the other. So the turning points are x = sqrt(3 lambda/20) and x = -sqrt(3 lambda/20).

If lambda >= 0 then dy/dx = 0 only at x = 0, which is not a turning point because dy/dx is positive on both sides of the value.

If lambda = -5/3 then the turning points are (1/2, -1/12), a minimum, and x = (-1/2, 1/12), a maximum. Please look at the attachment.
Attached Images

6. Was very helpful, thank you. Hope I can do similar ones which prolly will come up.

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