The least value of the function $$F(x)=$$∫$$0x$$ (3$$ $$sin$$ $$u+4$$ $$cos$$ $$u) duon the interval $$(5$$π$$/4$$, $$4$$π$$/3$$] is

Question

The least value of the function $$F(x)=$$∫$$0x$$   (3$$ $$sin$$ $$u+4$$ $$cos$$ $$u) duon the interval $$(5$$π$$/4$$, $$4$$π$$/3$$] is

Infinity Learn · Accepted Answer

We have F'(x)=3$$ $$sin$$ $$x+4$$ $$cos$$ x. Since $$sin$$ x and $$cos$$ x assume negative values in the third quadrant, we have F′$$(x)<$$0$$ for all x∈(5$$π$$/4$$, $$4$$π$$/3) so $$F(x)$$ assumes the least value at the point x $$=$$ $$4$$π$$/3$$.Thus the least value is $$F(4$$π$$/3)=$$∫$$04$$π$$/3$$ $$(3sin$$⁡ $$u+4cos$$⁡ $$u)du=($$−$$3cos$$⁡ $$u+4sin$$⁡ $$u)04$$π$$/3=$$−$$3cos$$⁡ $$4$$$$π$$$$3+4sin$$⁡ $$4$$$$π$$$$3$$−$$($$−$$3)=92$$−$$432=9$$−$$432$$.

The least value of the function $F (x) = \int_{0}^{x}$ $(3 \sin u + 4 \cos u) d u$
on the interval $(5 π / 4, 4 π / 3]$ is

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The least value of the function F(x)=∫0x (3 sin u+4 cos u) duon the interval (5π/4, 4π/3] is

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The least value of the function $F (x) = \int_{0}^{x}$ $(3 \sin u + 4 \cos u) d u$
on the interval $(5 π / 4, 4 π / 3]$ is