Question
Question: If \(F = \dfrac{{\mu mg}}{{\cos \theta + \mu \sin \theta }}\) , then \(F\) is minimum for A. \(\co...
If F=cosθ+μsinθμmg , then F is minimum for
A. cosθ=μ
B. sinθ=μ
C. tanθ=μ
D. cotθ=μ
Solution
We can find the minimum value of the given function by using the concept of maxima and minima. We first derivate the given function with respect to θ and then equate it to zero, to obtain the condition for minimum value of the function. Also, we calculate the minimum value of the function.
Complete step by step answer:
The given function is
F=cosθ+μsinθμmg
Here, μ,m,g are constants.
Differentiating the given equation with respect to θ, we get
dθdF=−μmg(cosθ+μsinθ1)2(−sinθ+μcosθ)
Now, For minima, we have to put dθdF=0, then
As μmg(cosθ+μsinθ1)2=0
So,
0=sinθ−μcosθ
⇒μcosθ=sinθ
We get, tanθ=μ
The value of function for tanθ=μ is
F=cosθ+tanθsinθtanθmg
∴F=sinθmg. This is the minimum value of the given function for tanθ=μ .
Hence, option C is correct.
Note: We should know the rules of differentiation i.e. dxd[f(x)1]=[(f(x))2−1]f′(x). We can also solve this question by putting the various options into the value of μ , but it will generate false results for every option, so we make use of differentiation.