let f be a differentiable function such that f(0)=-5 and f'(x) 3 for all x, but which value is not possible for f(2)?

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Guest Feb 6, 2016

#2**+10 **

let f be a differentiable function such that f(0)=-5 and f'(x)**= **3 for all x, but which value is not possible for f(2)?

what you must remember is that f'(x) is the gradient of the tangent to the curve. If this gradient is any constant then the curve MUST be a line.

SO

This is a line with a gradient ot 3 and a y intercept of -5

the equation has to be

f(x)=3x-5

f(2)=2*3-5 = 1

No other value is possible.

Melody
Feb 6, 2016

#1**+5 **

im guessing its gonna be 2 b/c maxmium increase is 6, and 2 is not in its range, is this right

Guest Feb 6, 2016

#2**+10 **

Best Answer

let f be a differentiable function such that f(0)=-5 and f'(x)**= **3 for all x, but which value is not possible for f(2)?

what you must remember is that f'(x) is the gradient of the tangent to the curve. If this gradient is any constant then the curve MUST be a line.

SO

This is a line with a gradient ot 3 and a y intercept of -5

the equation has to be

f(x)=3x-5

f(2)=2*3-5 = 1

No other value is possible.

Melody
Feb 6, 2016

#4**+5 **

ok then, if the gradient was equal to 3 then f(2)=1

if the gradient is less than 3 then f(2) must be less than 1 (because it is not so steep anywhere)

so f(2) cannot be more than 1. Only one of those points is more than 1. :)

Melody
Feb 6, 2016

#5**+10 **

Thanks, Melody.......!!! ......here's another way to see this...

This must be a linear equation if the derivative is just some integer at all x values...so....

At 0, f(x) = -5

So....at f(2).......x has changed by 2 units

Thus, using the formula for slope :

[ f(2) - (-5) ] / [ 2 - 0] <= 3 simplify

[ f(2) + 5] / 2 <= 3 multiply both sides by 2

f(2) + 5 <= 6 subtract 5 from both sides

f(2) <= 1

Thus f(2) cannot = 2 because this is * greater* than 1

CPhill
Feb 6, 2016