If θ is an angle in standard position and its terminal side passes through the point (7,-4), find the exact value of sec
⁡
θ
secθ in simplest radical form.

9514 1404 393

  (√65)/7

Step-by-step explanation:

We can use the relation between the secant and the tangent:

  sec(θ)² = tan(θ)² +1

  sec(θ) = √(1 + (-4/7)²) = √(65/49)

  sec(θ) = (√65)/7 . . . . . . secant is positive in the 4th quadrant

(a) Increasing in the interval (-623, 2) and (623, ∞)

f is decreasing in the interval (∞, -623) and (2, - 623).

(b) Local maximum is f = 2 .

2 local minima  f = -623.

Step-by-step explanation:

f(x) = x^4 - 50x^2 + 2

Finding the derivative:

f'(x) = 4x^3 - 100x

This = 0 at  a turning point:

4x(x^2 - 25) = 0

x = 0,  x + -5, 5.

Find the second derivative:

f"(x) = 12x^2 - 100

When x = 0 f(x) = 2

When x = 0, f"(x) is negative so the point (0, 2) is a maximum .

When x = -5, f"(x) is  positive . Also positive when x = 5.

So the 2 local minimums are (-5, (-5)^4 - 50(-5)^2 + 2) = (-5, -623) and

(5, -623).