Question

find the minimum value of 2^x - 4x +1

Answer 1

@tanjung Do you know calculus?

Answer 2

yea, but still stuck

Answer 3

First find the derivative of this.
Let this be y
\[y=2^x-4x+1\]
Find \(\frac{dy}{dx}\)

Answer 4

I want conform 2^x or x^2 ?

Answer 5

2^x not x^2

Answer 6

Yeah

Answer 7

from 2^x - 4x +1 you need to differentiate this there is a rule that the derivative of a^x = a^x * lna so let us apply this here to obtain

f'(x) = 2^x * ln2 -4 and because we are finding critical calues we will let our slope equal zero therefor now we have

2^x * ln2 -4=0 and now we simply just solve fo x which is 2^x = 4/ln2 and to solve for x we would have to take ln on both sides to get
ln2^x = 4/ln2 --> x*ln2 = 4/ln2 --> x = 4/(ln2)^2 :)

Answer 8

Not quite:
\[f(x)=2^x-4x+1\]
Solving for a local minimum:
\[f^{\prime}(x)=2^xln({2})-4=0\]
\[2^xln({2})=4\]
\[2^x=\frac{4}{ln({2})}\]
\[xln(2)=ln(\frac{4}{ln({2})})\]
\[x=\frac{ln(\frac{4}{ln({2})})}{ln(2)}\]

Answer 9

oo thanks i forgot to include ln on the right side

Answer 10

That's one way where LaTeX makes it a lot easier to spot mistakes.

Answer 11

hoho..im wrong in used differential of 2^x, ...
so, we need subtitute the value x (critical) to function original isn't it?

Answer 12

hey tanjung, are you here?

Answer 13

yeahhh

Answer 14

yeah, you have to substitute back the value of x in your original expression to find the minimum value

Answer 15

ya, igot it.. ymin=(4 - 7ln(2) +4*ln(ln(2))/ln(2)..
value y not simple yea..
.

Answer 16

Did you match it?

Answer 17

yeahh, correct m if i still wrong, please

Answer 18

You found that
\[x=\frac{ln(\frac{4}{ln({2})})}{ln(2)}\]
so it's
\[x=\log_2 (\frac{4}{\ln 2})\]
since
\[\frac{\ln a}{\ln b}=\log_b a\]
Now we have
\[2^x-4x+1\]
Let's substitute
\[2^{\log_2 (\frac{4}{\ln 2})}-4{\log_2 (\frac{4}{\ln 2})}+1\]
we know that
\[a^{\log_a b}=b\]
so
\[\frac{4}{\ln 2} -4\log_2 (\frac{4}{\ln 2})+1\]
Now you can simplify this, can't you?

Answer 19

oke, thank you ash... i understand know.

Answer 20

good:D

Answer 21

haahh, sorri... forget to say thank you for valpey to,,, very" sorri sir :)