Hi there! Can you help me with this MCQ question..
The largest value of x for which the function $\displaystyle f(x) = 3sin x + 4 cos x - 2$ is...
a) 7
b) 6
c) 5
d) 3
e) none of the above
And the ans in the ans key is 3
What you've posted doesn't make sense. You've just written a function definition. This is valid for all x and thus there is no largest value of x for which this is true.
Further, if you meant the largest value of x such that f(x)=C, where C is some constant, there is still no largest x because f(x) is periodic and will be equal to C infinitely many times as x grows without bound.
Do you mean what is the maximum value that f(x) attains?
The way to solve this is to note that
$\displaystyle A\cos(x)+B\sin(x)=\sqrt{A^2 + B^2}\cos(x + \phi)$ thus
$\displaystyle f(x)=\sqrt{3^2+4^2}\cos(x+\phi)-2$ (there is a formula for $\displaystyle \phi$ but it doesn't matter here)
This clearly attains a maximum value of $\displaystyle \sqrt{9+16}-2=3$
This matches your answer (d) and so is probably what is meant.
If this is really the way the question was worded I would have a chat w/your teacher. The question sucks.
We need to know what the problem really says! What you have "The largest value of x for which the function f(x) = 3sin x + 4 cos x - 2 is..." makes no sense. My first thought was that it asks for the largest value for which the function "does" something or has some value. But since f(x) is periodic and repeats I don't see how the can be such a largest value.
I strongly suspect that the problem asks for the largest value of f(x), not x. There are two very different ways to answer that. The simpler way requires Calculus: find a value of x that makes the derivative 0 and evaluate f(x) for that value of x. The other is to first drop the constant, "-2", and look only at "3 sin(x)+ 4 cos(x)". There exist numbers "A" and "b" such that Asin(x+ b)= 3sin(x)+ 4cos(x) for all x. Can you find them? Of course, in that case, since the maximum value of sin(x+ b) is 1, the maximum value of "3 sin(x)+ 4 cos(x)" is just "A". And the maximum value of 3 sin(x)+ 4 cos(x)- 2 is, of course, A- 2.