Menu Close

for-every-real-set-R-f-R-and-f-is-Smooth-function-and-f-is-f-C-2-x-f-1-x-gt-0-f-2-x-lt-0-then-prove-0-t-cos-f-x-dx-2-f-1-t-t-R-

Tinku Tara 0 Comments
Pin




Question Number 213290 by issac last updated on 02/Nov/24
for every real set R , f∈R and f is Smooth function. and f is f∈C^2 ∀_x f^((1)) (x)>0 , f^((2)) (x)<0 then prove ∣∫_0 ^( t) cos(f(x))dx∣≤(2/(f^((1)) (t))) t∈R
foreveryrealsetR,f∈RandfisSmoothfunction.andfisf∈C2∀xf(1)(x)>0,f(2)(x)<0thenprove∣∫0tcos(f(x))dx∣≤2f(1)(t)t∈R
Answered by MrGaster last updated on 02/Nov/24
∣∫_0 ^t cos(f(x))dx∣ =∣[((sin(f(x)))/(f′(x)))]_0 ^t −∫_0 ^t ((sin(f(x))∫^(′′) (x))/((f′(x))^2 ))dx∣ ≤∣((∣sin(f(t))∣)/(f′(t)))+((∣sin(f(0))∣)/(f′(0)))+∫_0 ^t ((∣sin(f(x))f′′(x))/((f′(x))^2 ))dx ≤((∣sin(f(t))∣)/(f′(t)))+((∣sin(f(0))∣)/(f′(0)))+∫_0 ^t ((∣sin(f(x))∣∣f′′(x)∣)/((f′(x))^2 ))dx ≤(1/(f′(t)))+(1/(f′(0)))+∫_0 ^t ((∣f^(′′) (x)∣)/((f′(x))^2 ))dx ≤(1/(f′(t)))+(1/(f′(0)))+∫_0 ^t ((−f′′(x))/((f′(x))^2 ))dx =(1/(f′(t)))+(1/(f′(0)))+[−(1/(f′(x)))]_0 ^t =(1/(f′(t)))+(1/(f′(0)))−(1/(f′(t)))+(1/(f′(0))) =(2/(f′(0))) ≤(2/(f′(t)))
∣∫0tcos(f(x))dx∣=∣[sin(f(x))f′(x)]0t−∫0tsin(f(x))∫″(x)(f′(x))2dx∣≤∣∣sin(f(t))∣f′(t)+∣sin(f(0))∣f′(0)+∫0t∣sin(f(x))f″(x)(f′(x))2dx≤∣sin(f(t))∣f′(t)+∣sin(f(0))∣f′(0)+∫0t∣sin(f(x))∣∣f″(x)∣(f′(x))2dx≤1f′(t)+1f′(0)+∫0t∣f″(x)∣(f′(x))2dx≤1f′(t)+1f′(0)+∫0t−f″(x)(f′(x))2dx=1f′(t)+1f′(0)+[−1f′(x)]0t=1f′(t)+1f′(0)−1f′(t)+1f′(0)=2f′(0)≤2f′(t)

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *