Mathematics: Use the mean value theorem to prove that if the derivative of f(x) is less than or equal to 2 for x 0 and f(0)=4 then f(x) is less than or equal to 2x+4 for all x greater than or equal to 0.

Use the mean value theorem to prove that if the

GLASGOWANASTASIA

09.01.2022

The proof is given below.

Step-by-step explanation:

Given:

$f'(x)\leq 2,\ f(0)=4, \ x0$

Using Mean Value Theorem,

$f'(c)=\frac{f(b)-f(a)}{b-a}$

Here, $a=0,\ b=x$

Therefore, $f'(c)=\frac{f(x)-f(0)}{x-0}=\frac{f(x)-4}{x}$

Now, as per question the derivative of f(x) is less than or equal to 2.

Therefore, $f'(c)\leq 2$ . Plug in the value of f'(c) . This gives,

$\frac{f(x)-4}{x}\leq 2\\f(x)-4\leq 2x ...............(\textrm{Cross product})\\f(x)\leq 2x+4 ..............(\textrm{Adding 4 both sides})$

Therefore, we get, $f(x)\leq 2x+4$ and hence it is proved using mean value theorem.

0,0(0 оценок)

vale2652

25.11.2020

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

here m = 7, hence y = 7x + c

To find c substitute ( 8, 6 ) into the equation

6 = 56 + c ⇒ c = 6 - 56 = - 50, thus

y = 7x - 50 → a

0,0(0 оценок)