is this statement true??? or false?

Question

a)Let a be a positive number not equal to 1 and let f(x)= a^x . Then f'(x)= a^x


b) The curve f(x)= 2x^3+4x+5 has no tangent line with slope 3.
c) If g is a differentiable function and k is a constant, then is differentiable, and

That is, the derivative of a function that is a product of a constant and a differentiable function is the constant times the derivative of the function.

which is true ? which is false? it is not a multiple question, each is seperate

Answer 1

1) F, since it's true only if a = e, Euler's constant, otherwise f'(x) = a^x Log(a)
2) T, since nowhere is f'(x) = 6x^2 + 4 is less than 4
3) T, if you mean that d/dx (a f(x)) = a d/dx (f(x)). Property of linearity.

But your posted question looks messed up, I don't think you have all of it.

Answer 2

a) False. Only true if a = e.
b) True. The slope is 6x² + 4 ≥ 4
c) True. But the question looks messed up.

Scythian has it right.

Answer 3

a. is false. The formal derivative of a^(cx) (for constant a and c) is d/dx a^cx=ca^(cx)*ln(a)

b. is true singe f'(x)=6x²+4 and a slope of 3 => 6x²+1=0 which has no real roots for x, thus nowhere an x value for a slope of 3.

c. is true since, if limit(σx -> 0) [g(x+σx)-g(x)]/σx exists (and is called g'(x)) then for arbitrary k we have
limit(σx ->0) [kg(x+σx)-kg(x)]/σx=(limit(σx ->0) kg'(x)

Hope that helps.


Doug

Answer 4

let a^x = e^y
y = x lna
f(x) = e^xlna
f'(x) =ln(a) e^xln(a) =ln(a)a^x
so a) is false

b) f'(x) = 6x^2 + 4 which cannot equal 3 for real x
so b) is true

If g(x) = kf(x) then g'(x) = kf'(x)
so c) is true

Answer 5

a and c are both true but b is false.

Answer 6

All the statements are true.

Answer 7

c