Question

Determine whether the statement is true or false ,and explain why or give an example that shows it is false

1) every nth-degree polynomial has(n-1) critical numbers.
2) The maximum value of y=3sinx+2cosx is 5
3) arcsin^2x+arccos^2x=1

Answer 1

4) If y=ax+b then \[\frac{ \Delta y}{ \Delta x }= \frac{ dy }{ dx }\]

Answer 2

@ganeshie8

Answer 3

first, lets look at the easy problem : #4

Answer 4

Notice that \(y = ax + b\) is in `slope-intercept` form :

\(\text{slope} = a = \dfrac{y_2-y_1}{x_2 - x_1} = \dfrac{\Delta y}{\Delta x}\)

Answer 5

by definition, derivative = slope
so \(\dfrac{\Delta y}{ \Delta x} = \dfrac{dy}{dx}\) is true whenever the function is a line.

Answer 6

ok

Answer 7

what about the other ones?

Answer 8

3) \(\arcsin^2x+\arccos^2x=1\)

just give a counter example when \(x = 0\)

Answer 9

\(\arcsin^2(0)+\arccos^2(0)= 0^2 + \left(\frac{\pi}{2}\right)^2 \ne 1 \)
Q.E.D.

Answer 10

in the second one i should find the maximum then put it value in the function to find y, right ?

Answer 11

you can do that, or simply use the below formula :

\(a\cos x + b\sin x = \sqrt{a^2+b^2} \cos (x - \alpha ) \)

Answer 12

\(\implies\) the max value of \(y=3\sin x+2\cos x \) is \(\sqrt{3^2 + 2^2}= \sqrt{13}\)
so the given statement is false

Answer 13

what about the first one ?

Answer 14

1) every nth-degree polynomial has(n-1) critical numbers.

strictly speaking its false; for ex : \(P(x) = x^4\) has only one critical number : x = 0

Answer 15

below is a correct statement however :
1) every nth-degree polynomial has `at most` (n-1) critical numbers.

Answer 16

@ganeshie8 thanks alot , and sorry for taking ur time :)

Answer 17

np.. . u wlc :)