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Hatshy [7]
3 years ago
13

Consider the function f(x) = 6sin(x^2) on the interval 0 ≤ x ≤ 3.

Mathematics
2 answers:
Vitek1552 [10]3 years ago
4 0
<span> a) F' = 6 sin(x^2) = 0
x^2 = pi
x = sqrt(pi)

b) Fmax = F(1) + integral [1, pi] f(x) dx = 9.7743 </span>
stira [4]3 years ago
4 0

Solution:

The given function is

f(x)= 6 sin x²→→→0 ≤ x ≤ 3.

Differentiating once

f'(x)= 6 cos x²× 2 x=12 x cos x²

For Maximum or Minimum

f'(x)=0

12 x cos x²=0

cos x²=0 ∧ x=0

cos x²=cos \frac{\pi }{2}

x²=  \frac{\pi}{2}

x= \pm \sqrt\frac{\pi}{2}

As, the interval given is,  0 ≤ x ≤ 3.

x=  \sqrt\frac{\pi}{2}= \frac{22}{14}

f"(x)=12 (-sin x²) × 2 x × x + 12 cos x²= 24 x². (-sin x²)+ 12 cos x²

At, x=  \sqrt\frac{\pi}{2}= \frac{22}{14}

f"(x)=-24 × \frac{\pi}{2} ×1 + 0= A negative number

Showing the function attains maxima at this point.

f(0)=6 sin 0=0

f(3)= 6 sin 3²= 6 sin 9(radian)=6 × 0.40(approx)= 2.40

f( \sqrt\frac{\pi}{2})=6 × sin(\frac{\pi}{2} )= 6 ×1=6→→Maximum Value

(b) f(1)=5

I.e at x=1, y=5

Maximum value attained by , f(x)=6 sin x² is 6 at , x=  \sqrt\frac{\pi}{2}.


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