11. find the absolute maximum and minimum of g(θ)=sin(θ) - θ on -π/2,π/2. 12. find the absolute maximum and…

Explanation:

Step1: Find the derivative of (g(\theta))

The derivative of (g(\theta)=\sin(\theta)-\theta) is (g'(\theta)=\cos(\theta) - 1).

Step2: Find the critical - points

Set (g'(\theta)=0), so (\cos(\theta)-1 = 0), which gives (\cos(\theta)=1). In the interval (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]), (\theta = 0) is the critical - point.

Step3: Evaluate (g(\theta)) at the critical - point and endpoints

Evaluate (g(\theta)) at (\theta=-\frac{\pi}{2}), (g\left(-\frac{\pi}{2}\right)=\sin\left(-\frac{\pi}{2}\right)+\frac{\pi}{2}=- 1+\frac{\pi}{2}\approx - 1 + 1.57=0.57); at (\theta = 0), (g(0)=\sin(0)-0 = 0); at (\theta=\frac{\pi}{2}), (g\left(\frac{\pi}{2}\right)=\sin\left(\frac{\pi}{2}\right)-\frac{\pi}{2}=1-\frac{\pi}{2}\approx1 - 1.57=-0.57).

Answer:

The absolute maximum value of (g(\theta)) on (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]) is (-1+\frac{\pi}{2}) at (\theta =-\frac{\pi}{2}), and the absolute minimum value is (1-\frac{\pi}{2}) at (\theta=\frac{\pi}{2}).

Explanation:

Step1: Find the derivative of (f(x))

The derivative of (f(x)=5x^{\frac{4}{5}}+4x) is (f'(x)=5\times\frac{4}{5}x^{-\frac{1}{5}}+4 = 4x^{-\frac{1}{5}}+4=\frac{4}{x^{\frac{1}{5}}}+4=\frac{4 + 4x^{\frac{1}{5}}}{x^{\frac{1}{5}}}).

Step2: Find the critical - points

Set (f'(x)=0), then (\frac{4 + 4x^{\frac{1}{5}}}{x^{\frac{1}{5}}}=0). The numerator (4 + 4x^{\frac{1}{5}}=0) gives (x^{\frac{1}{5}}=-1), so (x=-1). Also, (f'(x)) is undefined at (x = 0). So the critical - points are (x=-1) and (x = 0). But we are considering the interval ([-2,1]).

Step3: Evaluate (f(x)) at the critical - points and endpoints

Evaluate (f(x)) at (x=-2), (f(-2)=5\times(-2)^{\frac{4}{5}}+4\times(-2)=5\times2^{\frac{4}{5}}-8\approx5\times1.74 - 8=8.7 - 8 = 0.7); at (x = 0), (f(0)=0); at (x = 1), (f(1)=5\times1^{\frac{4}{5}}+4\times1=5 + 4=9).

Answer:

The absolute maximum value of (f(x)) on ([-2,1]) is (9) at (x = 1), and the absolute minimum value is (0) at (x = 0).