Find range of the functionf(x)=3|sin⁡x|−4|cos⁡x|I tried to do by using the trigonometric identitiessin2x=1−cos⁡2x2; cos2x=1+cos⁡2x2Sof(x)=31−cos⁡2x2−41+cos⁡2x2but don't know how to proceed from here

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Find range of the functionf(x)=3|sin⁡x|−4|cos⁡x|I tried to do by using the trigonometric identitiessin2x=1−cos⁡2x2;  cos2x=1+cos⁡2x2Sof(x)=31−cos⁡2x2−41+cos⁡2x2but don&#039;t know how to proceed from here

Kayla Kline · Accepted Answer

f(0)=−4,  f(π2)=3 and   −4≤f(x)≤3,   for all x∈R. Intermediate value theorem gives f(R)=[−4,3].

Mollie Nash · Accepted Answer

Letf(x)=3|sin⁡x|−4|cos⁡x|It&#039;s easy to show f(x) is periodic with T=π and f(x) is even. So it&#039;s enough to consider 0≤x≤π2If 0≤x≤π2 then0≤sin⁡x≤1⇒0≤|sin⁡x|≤1And0≤cos⁡x≤1⇒0≤|cos⁡x|≤1Therefore we have −4≤f(x)≤3 but this could be only an upper bound not the actual range.If 0≤x≤π2 then(x)=3sin⁡x−4cos⁡x⇒f′(x)=3cos⁡x+4sin⁡x&amp;gt;0⇒f(0)≤f(x)≤f(π2)⇒−4≤f(x)≤3Confirming the first result. Actually because f(x) is an increasing function in that interval, the results are the same. So we can conclude that −4≤f(x)≤3 and WA verifies that.

Vasquez · Accepted Answer

f(x)=3|sin⁡x|−4|cos⁡x|−4≤−4|cos⁡x|≤f(x)≤3|sin⁡x|≤3By inspection:f(π2)=3;f(0)=−4Rangef=[−4,3]

Find range of the function f(x)=3 |\sin x|-4|\cos x| I tried to

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